On higher-order discriminants
Vladimir Petrov Kostov
 
Synergistic Team Composition
    
Ewa Andrejczuk

       Artificial Intelligence Research Institute (IIIA-CSIC)

       Change Management Tool S.L

       Barcelona, Spain

       ewa@iiia.csic.es
Juan A. Rodríguez-Aguilar 

       Artificial Intelligence Research Institute (IIIA-CSIC)

       Barcelona, Spain

       jar@iiia.csic.es
Carme Roig

       Institut Torras i Bages

       L'Hospitalet de Llobregat, Spain

       mroig112@xtec.cat
Carles Sierra

       Artificial Intelligence Research Institute (IIIA-CSIC)

       Barcelona, Spain

       sierra@iiia.csic.es

    December 30, 2023
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For the family of polynomials in one variable 
P:=x^n+a_1x^n-1+⋯ +a_n, n≥ 4, we consider its 
higher-order discriminant sets {D̃_m=0}, where 
D̃_m:=Res(P,P^(m)), 
m=2, …, n-2, and their projections in the spaces of 
the variables a^k:=(a_1,… ,a_k-1,a_k+1,… ,a_n). 
Set P^(m):=∑ _j=0^n-mc_ja_jx^n-m-j, P_m,k:=c_kP-x^mP^(m). 
We show that Res(D̃_m,∂D̃_m/∂ a_k,a_k)=
A_m,kB_m,kC_m,k^2, where A_m,k=a_n^n-m-k, 
B_m,k=Res(P_m,k,P_m,k') 
if 1≤ k≤ n-m and A_m,k=a_n-m^n-k, 
B_m,k=Res(P^(m),P^(m+1)) if n-m+1≤ k≤ n.
The equation C_m,k=0 defines the projection in the space 
of the variables a^k 
of the closure of the set of values of (a_1,… ,a_n) 
for which P and P^(m) have two 
distinct roots in common. The polynomials B_m,k,C_m,k∈ℂ[a^k] 
are irreducible. The result is generalized to the case when P^(m) 
is replaced by a polynomial P_*:=∑ _j=0^n-mb_ja_jx^n-m-j, 
0≠ b_i≠ b_j≠ 0 for i≠ j.
 

AMS classification: 12E05; 12D05


Key words: polynomial in one variable; discriminant set; 
resultant; multiple root




§ INTRODUCTION


In this paper we consider for n≥ 4 the general family of monic polynomials 
in one variable 
P(x,a):=x^n+a_1x^n-1+⋯ +a_n, x,a_j∈ℂ. For its mth 
derivative w.r.t. x we set 
P^(m):=c_0x^n-m+c_1a_1x^n-m-1+⋯ +c_n-ma_n-m, 
where c_j=(n-j)!/(n-m-j)!. 
For m=1, …, 
n-1 we define the mth order discriminant of P as 
D̃_m:=Res(P,P^(m)) which is the determinant of the Sylvester matrix 
S(P, P^(m)). We remind that S(P, P^(m)) is 
(2n-m)× (2n-m), its first (resp. (n-m+1)st) row equals 


    (1,a_1,… ,a_n,0,… ,0)      (resp.      
    (c_0,c_1a_1,… ,c_n-ma_n-m,0,… ,0)    )  ,

the second (resp. (n-m+2)nd) row is obtained from this one by shifting 
by one position to the right and by adding 0 to the left etc. We say that 
the variable a_j is of quasi-homogeneous weight j because up to 
a sign it equals the jth elementary symmetric polynomial in the roots of 
the polynomial P; the quasi-homogeneous weight of x is 1.  

There are at least two problems in which such discriminants are of interest. 
One of 
them is the Casas-Alvero conjecture that if a complex univariate polynomial has 
a root in common with each of its nonconstant derivatives, then it is a power 
of a linear polynomial, see <cit.>, <cit.> and <cit.> 
and the claim in <cit.> that the answer to 
the conjecture is positive. 

Another one is the study of the 
possible arrangements of the roots of a hyperbolic polynomial (i.e. real and 
with all roots real) and of all its nonconstant derivatives on the real line. 
This problem can be generalized to a class of polynomial-like functions 
characterized by the property their nth derivative to vanish nowhere. 
It turns out that for this class Rolle's theorem gives only necessary, 
but not sufficient 
conditions for realizability of a given arrangement by the zeros of 
a polynomial-like function, see <cit.>, <cit.>, <cit.> 
and <cit.>. Pictures 
of discriminants for the cases n=4 and n=5 can be found in <cit.>. 
Properties of the discriminant set {D̃_1=0} for real polynomials 
are proved in <cit.>. 

A closely related question to the one of the 
arrangement of the roots of a hyperbolic polynomial is the one 
to study overdetermined strata in the space of the coefficients of the 
family of polynomials P (the definition is given by B. Z. Shapiro 
in <cit.>); these are sets of values of the coefficients for which there 
are more equalities between roots of the polynomial and its derivatives than 
expected. Example: the family of polynomials x^4+ax^3+bx^2+cx+d depends on 
4 parameters two of which can be eliminated by shifting and rescaling the 
variable x which gives (up to a nonzero constant factor) the family 
S:=x^4-x^2+cx+d. For c=0, d=1/2 the polynomial has two double roots 
± 1/√(2), and 0 is a common root for S' and S”'. This makes 
three independent equalities, i.e. more than the number of parameters. For 
polynomials of small degree, overdetermined strata have been studied in 
<cit.> and <cit.>. The study of overdetermined strata is interesting 
both in the case of complex and in the case of real coefficients. 

In what follows we enlarge the context by considering instead of the 
couple of polynomials (P,P^(m)) the couple (P,P_*), where 
P_*:=∑ _j=0^n-mb_ja_jx^n-m-j, b_j≠ 0 and b_i≠ b_j 
for i≠ j. By abuse of notation we set D̃_m:=Res(P,P_*). 



The polynomial D̃_m is irreducible. It is a degree n polynomial 
in each of the variables a_j, j=1, …, n-m, and a degree 
n-m polynomial in each of the variables a_j, j=n-m+1, …, n. 
It contains monomials M_j:=± b_j^na_j^n(1-b_0/b_j)^ja_n^n-m-j, 
j=1, …, n-m, and 
N_s:=± b_n-m^m-sa_n-m^m-sb_0^n-m+sa_n-m+s^n-m, s=1, …, m-1. 
It is quasi-homogeneous, of 
quasi-homogeneous weight n(n-m). The monomial M_j (resp. N_s) is 
the only monomial containing a_j^n (resp. a_n-m+s^n-m).



We prove first the presence in D̃_m of the monomials M_j and N_s. 
For each j fixed, 1≤ j≤ n-m, one can subtract the (n-m+ν )th row 
of S(P,P_*) multiplied by 1/b_j from its νth one, 
ν =1, …, n-m. We denote by T the new matrix.
One has T= S(P,P_*) and  
the variable a_j is not present in the first n-m rows of T. Thus there 
remains a single term of T containing n factors a_j; it is 
obtained when the entries b_ja_j in positions (n-m+μ ,j+μ ) of T, 
μ =1, …, n, are multiplied by the entries a_n in positions 
(ℓ ,n+ℓ ), ℓ =j+1, …, n-m, 
and by the entries 1-b_0/b_j in 
positions (ℓ ,ℓ ), ℓ =1, …, j; this gives the monomial 
M_j. (If when computing S(P,P_*) 
one chooses to multiply the n entries b_ja_j, then they must 
be multiplied by entries of the matrix obtained from S(P,P_*) by deleting 
the rows and columns of the entries b_ja_j. This matrix is block-diagonal, 
its upper left block is upper-triangular, with diagonal entries equal to 
1-b_0/b_j, its right lower block is lower-diagonal, with diagonal 
entries equal to a_n. Hence M_j is the only monomial containing n 
factors a_j.)

To obtain the monomial N_s one chooses in the definition of T 
above j=n-m. Hence the first n-m 
rows of T do not contain the variable a_n-m. The monomial N_s is 
obtained by multiplying the entries a_n-m+s in positions 
(r,n-m+s+r), r=1, …, n-m, by the entries b_n-ma_n-m in 
positions (q,q), q=2n-2m+s+1, …, 2n-m and by the entries b_0 
in positions (n-m+p,p), p=1, …, n-m+s. The monomial N_s 
is the only one containing n-m factors a_n-m+s (proved by analogy with 
the similar claim about the monomial M_j).

The matrix S(P,P_*) contains each of the variables a_j, j=1, …, 
n-m (resp. a_s, s=n-m+1, …, n) 
in exactly n (resp. n-m) of its columns. 
The presence of the monomials M_j (resp. N_s) in D̃_m shows that 
D̃_m is a degree n polynomial in the variables a_j and a degree 
n-m one in the variables a_s. 

Quasi-homogeneity of D̃_m follows from the fact that its zero set 
and the zero sets of the polynomials P and P_* remain 
invariant under the quasi-homogeneous dilatations x↦ tx, 
a_κ↦ t^κa_κ, κ =1, …, n. 
Each of the monomials M_j and N_s is of quasi-homogeneous weight n(n-m). 

Irreducibility of D̃_m results from the impossibility to present 
simultaneously all monomials M_j and N_s  
as products of two monomials, 
of quasi-homogeneous weights u and n(n-m)-u, 
for any 1≤ u≤ n(n-m)-1. 



For Q,R∈ℂ[x] we denote by Res(Q,R) the resultant of Q 
and R and we write P^(m) for d^mP/dx^m. 
This refers also to the case when the coefficients of Q and R 
depend on parameters. We set a:=(a_1,… ,a_n) (resp. 
a^j=(a_1,… ,a_j-1,a_j+1,… ,a_n)) and we denote by 
𝒜≃ℂ^n (resp. 𝒜^j≃ℂ^n-1) 
the space of the variables a (resp. a^j). 
For K,L∈ℂ[a] we write S(K,L,a_k) and Res(K,L,a_k) 
for the Sylvester matrix and the resultant of K and L 
when considered as polynomials in a_k. We set 
D̃_m,k:=Res(D̃_m,∂D̃_m/∂ a_k,a_k). 
For a matrix A we denote by A_k,ℓ its entry in position (k,ℓ ) 
and by [A]_k,ℓ the matrix obtained from A 
by deleting its kth row and ℓth column. By Ω 
(indexed, with accent or not) we denote throughout the paper 
nonspecified nonzero constants. 
By P_m,k (1≤ k≤ n-m) we denote the polynomial b_kP-x^mP_*;  
its coefficients of x^n and x^k equal b_k-b_0≠ 0 and 0.
 


For 1≤ m≤ n-2 we denote by Θ and M̃ 
the subsets of the hypersurface 
{D̃_m=0}⊂𝒜 such that for a∈Θ (resp. 
for a∈M̃) the polynomial P has a root which is a double 
root of P_* (resp. the polynomials P and P_* have two 
simple roots in common). The remaining roots of P and P_*  
are presumed 
simple and mutually distinct. We call the set M̃ 
the Maxwell stratum of {D̃_m=0}.   


In the present paper we prove the following theorem;


Suppose that 2≤ m≤ n-2. Then:

(1) The polynomial D̃_m,k can be represented in the form 


    D̃_m,k=A_m,kB_m,kC_m,k^2 ,
 
where A_m,k=a_n^n-m-k if k=1, …, n-m, and 
A_m,k=a_n-m^n-k if k=n-m+1, …, n, 
B_m,k and C_m,k are irreducible 
polynomials in the variables a^k.

(2) One has B_m,k=Res(P_m,k,P_m,k') if k=1, …, n-m, and 
B_m,k=Res(P_*,P_*') if k=n-m+1, …, n.

(3) The equation C_m,k=0 defines the projection in the space 𝒜^k 
of the closure of the Maxwell stratum.



The paper is structured as follows. After some examples and remarks in 
Section <ref>, we justify in  
Section <ref> the form of the factor A_m,k, see 
Proposition <ref>; Section <ref> begins with Lemma <ref> 
which gives the form of the determinant of 
certain matrices that appear in the proof of Theorem <ref>. 
Section <ref> contains  
Lemma <ref> and Statements <ref>, <ref> 
and <ref> (the latter claims that the factors B_m,k and C_m,k 
are irreducible). They  
imply that one has 
D̃_m,k=A_m,kB_m,k^s_m,kC_m,k^r_m,k, where s_m,k, r_m,k 
∈ℕ, see 
Remark <ref>. Thus after Section <ref> there 
remains to show only that 
s_m,k=1 and r_m,k=2. 
In Section <ref> we prove Theorem <ref> 
in the case m=n-2, see Proposition <ref>. In Section <ref> 
we show that s_m,k=1. We finish the proof 
of Theorem <ref> in Section <ref>, by induction on n 
and m, as follows. Statement <ref> deduces formula (<ref>) 
for n=n_0+1, k=k_0+1 from formula (<ref>) 
for n=n_0, k=k_0. Statement <ref> justifies formula (<ref>) 
for n=n_0, 2≤ m<n_0-2, k=1 
using formula (<ref>) for n=n_0, m=n_0-2, k=1 
(recall that the latter is justified in Section <ref>). 


Acknowledgement. The author is deeply grateful to B. Z. Shapiro from 
the University of Stockholm for having 
pointed out to him the importance to study discriminants 
and for the fruitful discussions of this subject.




§ EXAMPLES AND REMARKS



Although Theorem <ref> speaks about the case 2≤ m≤ n-2,  
our first example treats 
the case m=1 in order to show its differences with the case 
2≤ m≤ n-2:
  

For n=3, m=1 we set P:=x^3+ax^2+bx+c, P_*:=x^2+Aax+Bb, 
0≠ A,B≠ 1, A≠ B. Then 


    [                                                                      D̃_1                                                                         =             (1-A)B(B-A)a^2b^2+(3AB-A-2B)abc+c^2+
    A^2(1-A)a^3c+B(1-B)^2b^3;                                                                          ;                                                                    D̃_1,1                                                                         = -A^2(A-1)^2  c  (-27A^2(1-A)c^2+4(A-B)^3b^3)  
    (-Ac^2+(1-A)B^2(1-B)b^3)^2;                                                                          ;                                                                    D̃_1,2                                                                         =      -B^2(B-1)^2  c  (-27B(1-B)^2c+4(A-B)^3a^3)  (-(1-B)c+A(1-A)^2Ba^3)^2;                                                                          ;                                                                    D̃_1,3                                                                         =                                    -(-4Bb+A^2a^2)  ((1-B)b-A(1-A)a^2)^2 . ]

The condition P and P_* to have two roots in common is 
tantamount to P_* dividing P. One has P=(x+a(1-A))P_*+W_1x+W_0, where 

    W_1:=(1-B)b-A(1-A)a^2  ,  W_0:=c-B(1-A)ab .

The quadratic factors in the above presentations of D̃_1,k, 
k=1, 2 and 3, are obtained by eliminating respectively a, b and c 
from the system of equations W_1=W_0=0 which is the necessary and sufficient 
condition P_* to divide P.

In the particular case A=2/3, B=1/3 (i.e. P_*=P'/3) one obtains

    D̃_1,1=(-2^6/3^15)  c  (-27c^2+b^3)^3  ,  D̃_1,2=(-2^6/3^15)  c  (-27c+a^3)^3  ,  D̃_1,3=(2^4/3^6)  (3b-a^2)^3 .










(1) For n≥ 4, m=1 and P_*=P' 
a result similar to Theorem <ref> holds 
true. Namely, if n≥ 4, then D̃_1,k 
is of the form A_1,kB_1,k^3C_1,k^2, where for m=1 the polynomials 
B_m,k and C_m,k are defined in the same way as for 2≤ m≤ n-2 
(with P_*=P'), but  
A_1,k=a_n^min (1,n-k)+max (0,n-k-2), see <cit.> and <cit.>. 
Hence for m=1 and P_*=P^(m) 
there are two differences w.r.t. the case m≥ 2 – the degree 3 
(instead of 1) of 
B_1,k, and A_1,n-1=a_n (instead of A_1,n-1=1). This difference can be 
assumed to stem 
from the fact that for m=1, if P has a root of multiplicity ≥ 3, then 
this is a root of multiplicity ≥ 2 for P'. This explanation is 
detailed below and in Remark <ref>. 




For n=4 and for generic values of b_j the polynomials 
D̃_1,k, up to a constant nonzero factor, are of the form 


    [                                         D̃_1,1                                              = (b_1/b_0)^3(1-b_1/b_0)^2  a_4^2 B̃_1,1C̃_1,1^2                                              ,                                         D̃_1,2                                              =  -(b_2/b_0)^2(1-b_2/b_0)^2  a_4 B̃_1,2C̃_1,2^2                                              ,;                                               ;                                         D̃_1,3                                              =  -(b_3/b_0)^2(1-b_3/b_0)^3  a_4 B̃_1,3C̃_1,3^2                                              ,                                         D̃_1,4                                              =                                 B̃_1,4C̃_1,4^2                                              , ]

where the polynomials B̃_1,k and C̃_1,k, when 
considered as polynomials in the variables a_j and b_j, are irreducible. 
Set b_1=3b_0/4, b_2=b_0/2, b_3=b_0/4. This is the case P_*=P'; we 
write B̃_1,k|_b_1=3b_0/4, b_2=b_0/2, b_3=b_0/4=B_1,k and 
C̃_1,k|_b_1=3b_0/4, b_2=b_0/2, b_3=b_0/4=C_1,k. In this case 
the polynomials C̃_1,k become reducible; they equal 
B_1,kC_1,k which explains the presence of the cubic factor B_1,k^3.

Thus for m=1 the genericity condition 
0≠ b_j≠ b_i≠ 0 (which we assume to hold true 
in the formulation of Theorem <ref>) is not sufficient in order 
to have the presentation (<ref>) for D̃_m,k. At the same 
time imposing a more restrictive condition means leaving outside the 
most interesting case P_*=P'.

(2) For m=n-1 
the analog of the factor C_m,k does not exist because P_* has a single 
root -b_1/b_0. For P_*=P^(n-1):=n!(x+a_1/n) this is x=-a_1/n. 
In this case one finds that 
D̃_n-1=(-1)^n(n!)^nP(-a_1/n). 
To see this one subtracts 
for j=1, …, n the jth column of the Sylvester matrix 
S(P,x+a_1/n) multiplied by -a_1/n from its (j+1)st column. This yields 
an (n+1)× (n+1)-matrix W whose entry in position (1,n+1) equals 
P(-a_1/n) and which below the first row has units in positions 
(ν +1,ν ), ν =1, …, n, and zeros elsewhere. 
Hence W=(-1)^nP(-a_1/n). There remains to remind that 
D̃_n-1= S(P,n!(x+a_1/n))=(n!)^n W.

One finds directly that 
D̃_n-1,k=∂D̃_n-1/∂ a_k=
(-1)^n(n!)^n(-a_1/n)^n-k, 
2≤ k≤ n. To find also D̃_n-1,1 one first observes that 

    P_n-1,1(x)/(n-1)!=-(n-1)x^n+a_2x^n-2+a_3x^n-3+⋯ +a_n
 
and that 
P(-a_1/n)=P_n-1,1(-a_1/n)/(n-1)!. Hence up to a nonzero rational factor the 
determinants of the matrices S(P_n-1,1,P_n-1,1') and 
S(D̃_n-1,∂D̃_n-1/∂ a_1,a_1) coincide, i.e. 
D̃_n-1,1=ĉRes(P_n-1,1,P_n-1,1'), 
ĉ∈ℚ.

(3) The fact that the factor C_m,k is squared (see formula (<ref>)) 
is not astonishing. At a generic point of the Maxwell stratum the hypersurface 
{D̃_m=0}⊂𝒜 is locally the intersection 
of two analytic hypersurfaces, see Statement <ref>. Consider a point 
Ψ∈𝒜^k close to the projection Λ _0 
in 𝒜^k of a generic 
point Λ∈M̃. There exist two points 
K_j∈{D̃_m=0}, 
j=1, 2, which belong to these hypersurfaces and are close to Λ, and 
whose common projection in 𝒜^k 
is Ψ.  
There exists 
a loop γ⊂𝒜^k, Ψ∈γ, 
which  circumvents the projection in 𝒜^k 
of the set Θ∪M̃ such that if one follows 
the two liftings on {D̃_m=0} of the points of γ which 
at Ψ are the points K_j, then upon one tour along γ these 
liftings are exchanged. Hence in order to define the projection of M̃ in 
𝒜^k by the zeros of an analytic function one has to 
eliminate this monodromy of rank 2 by taking the square of C_m,k. For 
the case m=1 a detailed construction of such a path γ is given 
in <cit.>.
   







For n=4 we consider the case of real polynomials. 
We write P=x^4+ax^3+bx^2+cx+d and we limit ourselves to 
the situation when P_*:=P^(m). 
On Fig. <ref> we show the sets {D̃_1=0} |_a=0 
and {D̃_2=0} |_a=0 when b, c and d are real. 
The sets {D̃_1=0} |_a=0 and {D̃_2=0} |_a=0 
are invariant under the 
quasi-homogeneous dilatations a↦ ta, b↦ t^2b, c↦ t^3c, 
d↦ t^4d, therefore the intersections of the sets with the subspaces 
{ b=0} and { b=± 1} give a sufficient idea about them. For each of 
these three intersections we represent the axes c and d, see Fig. <ref>. 
For b=-1 the set {D̃_1=0} |_a=0 is a curve with one 
self-intersection point at S and two ordinary 2/3-cusps at U and V; 
it is drawn in solid line. At U and V the polynomial P 
has one triple and one simple real root. The set 
{D̃_2=0} |_a=0,b=-1 consists of two 
straight (dashed) lines intersecting at H and tangent to the set 
{D̃_1=0} |_a=0 at the cusps U and V.

The sets {D̃_1=0} |_a=0,b=0 and {D̃_1=0} |_a=0,b=1 are 
parabola-like curves, the former has a 4/3-singularity at the origin while 
the latter is smooth everywhere. 
The set {D̃_1=0} |_a=0,b=1 contains 
an isolated double point T. 
The set {D̃_2=0} |_a=0,b=0 (resp. 
{D̃_2=0} |_a=0,b=1) is the c-axis (resp. the point L). 
The points S, T and for b=0 the origin 
belong to a parabola (because the quasi-homogeneous weights of the 
variables a_2 and a_4 equal 2 and 4 respectively). 
So do the points H, L 
and the origin for b=0. At S (resp. T) the polynomial P has two 
real (resp. two imaginary conjugate) double roots. At H and L the 
polynomial P is divisible by P”.  

Globally the set {D̃_2=0} |_a=0 is diffeomorphic 
to a Whitney umbrella. The set {D̃_2=0} |_a=0 is smooth 
along the c-axis for b=0 (except at the origin) 
and its tangent plane is the cd-plane.
  







§ THE FACTOR A_M,K


The following lemma will be used in several places of this paper:


Consider a p× p-matrix A having nonzero entries only 

i) on the diagonal (denoted by r_j, 
in positions (j,j), j=1, …, p);  

ii) in positions 
(ν ,ν +s), ν =1, …, p-s (denoted by q_ν), 
1≤ s≤ p-1, and 

iii) in positions (μ +p-s,μ ), μ =1, …, s (denoted by 
q_μ +p-s). 

Then A=r_1⋯ r_p± q_1⋯ q_p.



Developing A w.r.t. its first row one obtains the equality


    A=r_1 B+(-1)^s+2q_1 C  ,   where     
    B=[A]_1,1     and     C=[A]_1,s+1 .

The matrix B contains p-1 entries r_j (namely, r_2, …, r_p) 
and p-2 entries q_ν (the ones with 1≠ν≠ 1+p-s). In the same 
way, the matrix C contains p-2 entries r_j (1≠ j≠ s+1) 
and p-1 entries q_ν (ν≠ 1). 

When finding B one can develop it w.r.t. 
that row or column in which there is an entry r_j and there is no entry 
q_ν. By doing so p-1 times one finds that 
B=r_2⋯ r_p. The + sign of this product follows from 
the entries r_j being situated on the diagonal. When finding C one can 
develop it w.r.t. that row or column in which there is an entry q_ν 
and there is no entry r_j. By doing so p-1 times one finds that 
C=± q_2⋯ q_p which proves the lemma.


In the present section we prove the following proposition:


(1) For k=n-m+1, …, n, the polynomial D̃_m,k is 
not divisible by any of the variables a_j, j≠ n-m.

(2) For k=1,… ,n-m, the polynomial D̃_m,k is 
not divisible by any of the variables a_j, j≠ n.

(3) For k=1, …, n-m, the polynomial D̃_m,k is divisible by 
a_n^n-m-k and not divisible by a_n^n-m-k+1.

(4) For k=n-m+1, … ,n, it is divisible by 
a_n-m^n-k and not divisible by a_n-m^n-k+1. 








We show first that for a_i=0, n-m≠ i≠ k, the polynomial D̃_m 
is of the form Ω 'a_n-m^n+Ω”a_n-m^n-ka_k^n-m. 
Indeed, in this case one can list 
the nonzero entries of the 
(2n-m)× (2n-m)-matrix S(P,P_*) 
and the positions in which they are situated: 


    [               1           (j,j)               ,           a_n-m       (j,j+n-m)               ,;                ;             a_k         (j,j+k)               ,    j=1,… ,n-m ,;                ;             b_0     (ν +n-m,ν )               ,      b_n-ma_n-m (ν +n-m,ν +n-m)               ,     ν =1,… ,n . ]
 
Subtract the (μ +n-m)th row multiplied by 1/b_n-m from the μth 
one for μ =1, …, n-m. This makes disappear the terms 
a_n-m in 
positions (j,j+n-m) while the terms 1 in positions (j,j) become 
equal to Ω _*:=1-b_0/b_n-m. 
The determinant of the matrix doesn't change. We 
denote the new matrix by T. 
To compute T one can develop it n-k times 
w.r.t. the last column; each time one has a single nonzero entry 
in this column, this is b_n-ma_n-m in  
position (2n-m-ℓ ,2n-m-ℓ ), ℓ =0, …, n-k-1.
The matrix T_1 which remains after deleting 
the last n-k rows and columns of 
T has the following nonzero entries, in the following positions:


    [            Ω _*           (j,j)               ,             a_k         (j,j+k)               ,    j=1,… ,n-m ,;                ;             b_0     (ν +n-m,ν )               ,      b_n-ma_n-m (ν +n-m,ν +n-m)               ,     ν =1,… ,k . ]
 
Clearly T=(b_n-ma_n-m)^n-k T_1. 
On the other hand the matrix T_1 satisfies the conditions of 
Lemma <ref> with p=n-m+k and s=k. Hence 
T_1=Ω̃'a_n-m^k+Ω̃”a_k^n-m and  
D̃_m=Ω 'a_n-m^n+Ω”a_n-m^n-ka_k^n-m.

But then the (2n-2m-1)× (2n-2m-1)-Sylvester matrix 
S^*:=S(D̃_m,∂D̃_m/∂ a_k,a_k) 
has only the following 
nonzero entries, in the following positions:


    [      Ω”a_n-m^n-k            (j,j)                ,       Ω 'a_n-m^n        (j,j+n-m)                ,   j=1,… ,n-m-1 ,;                 ; (n-m)Ω”a_n-m^n-k    (ν +n-m-1,ν )                ,                ,    ν =1… ,n-m  . ]

Part (1) follows from  
S^*=
± (Ω 'a_n-m^n)^n-m-1((n-m)Ω”a_n-m^n-k)^n-m≢0.








We prove that for a_i=0, k≠ i≠ n, the polynomial D̃_m 
is of the form Ω ^†a_n^n-m+Ω ^††a_n^n-m-ka_k^n. 
Indeed, we list below the nonzero entries of the matrix S(P,P_*) 
and their positions:


    [             1         (j,j)             ,                         a_k       (j,j+k)                           ,;              ;           a_n       (j,j+n)             ,  j=1,… ,n-m ,;              ;           b_0   (ν +n-m,ν )             ,        b_ka_k (ν +n-m,ν +k)             ,   ν =1,… ,n . ]
 
One can develop n-m-k times S(P,P_*) w.r.t. its last column, 
where the only nonzero entries equal a_n. Thus 
S(P,P_*)=± a_n^n-m-k H, where H is obtained from 
S(P,P_*) by deleting the last n-m-k columns and the rows with indices 
k+1, …, n-m. The matrix H has the following nonzero entries, in 
the following positions:


    [           1       (j,j)           ,                     a_k     (j,j+k)           ,;            ;         a_n     (j,j+n)                       ,  j=1,… ,k ,;            ;         b_0   (ν +k,ν )           ,      b_ka_k (ν +k,ν +k)           , ν =1,… ,n . ]

For μ =1, …, k one can subtract the (μ +k)th row multiplied 
by 1/b_k from the μth one to make disappear the terms a_k in 
positions (μ ,μ +k); the entries 1 in positions (μ ,μ ) change 
to Ω ^*:=1-b_0/b_k. We denote the newly obtained matrix by H_1. 
Obviously 
H_1= H; we list the nonzero entries of H_1 and their respective 
positions:


    [        Ω ^*       (j,j)           ,                     a_n     (j,j+n)                       ,  j=1,… ,k ,;            ;         b_0   (ν +k,ν )           ,      b_ka_k (ν +k,ν +k)           , ν =1,… ,n . ]

One applies Lemma <ref> with p=n+k, s=n 
to the matrix H_1 to conclude that 
H= H_1=(Ω ^*)^k(b_ka_k)^n± b_0^na_n^k, so 
D̃_m= S(P,P_*)=
Ω ^†a_n^n-m+Ω ^††a_n^n-m-ka_k^n. 

But then the (2n-1)× (2n-1)-Sylvester matrix 
S(D̃,∂D̃/∂ a_k,a_k) has only the following 
nonzero entries, in the following positions:


    [  Ω ^††a_n^n-m-k           (j,j)               ,                     Ω ^†a_n^n-m         (j,j+n)                               ,    j=1,… ,n-1 ,;                ; nΩ ^††a_n^n-m-k     (ν +n-1,ν )               ,     ν =1,… ,n . ]

Its determinant equals 
± (Ω ^†a_n^n-m)^n-1(nΩ ^††a_n^n-m-k)^n
≢0 which proves part (2).




For k=1, … ,n-m the polynomial 
D̃_m contains the monomial 
M_k:=± b_k^na_k^n(1-b_0/b_k)^ka_n^n-m-k, and it  
does not contain any other monomial of 
the form Ω a_k^nE, where E is a product of powers 
of variables a_i with i≠ k, see Proposition <ref>.  

















Hence the first column of the (2n-1)× (2n-1)-matrix 
Y:=S(D̃_m,∂D̃_m/∂ a_k,a_k) contains only two 
nonzero entries, and these are Y_1,1=± b_k^n(1-b_0/b_k)^ka_n^n-m-k and 
Y_n,1=± nb_k^n(1-b_0/b_k)^ka_n^n-m-k. Thus 
Δ := Y  
is divisible by a_n^n-m-k. We consider two cases:

Case 1: k=n-m. We have to prove that D̃_m,n-m|_a_n=0≢0. 
Set a_j=0 for n-m≠ j≠ n-1. Hence the nonzero entries of the matrix 
S(P,P_*) and their positions are 


    [               1           (j,j)               ,           a_n-m       (j,j+n-m)               ,;                ;           a_n-1       (j,j+n-1)               ,    j=1,… ,n-m ,;                ;             b_0     (ν +n-m,ν )               ,      b_n-ma_n-m (ν +n-m,ν +n-m)               ,     ν =1,… ,n . ]

One can subtract the (j+n-m)th row multiplied by 1/b_n-m from the 
jth one, j=1, …, n-m, to make disappear the terms a_n-m 
in the first n-m rows. This doesn't change S(P,P_*). The terms 
1 in positions (j,j) are replaced by 1-b_0/b_n-m. 
Hence D̃_m is of the form  
Ω _1a_n-m^n+Ω _2a_n-1^n-ma_n-m (one first develops 
S(P,P_*) w.r.t. the last column, where there is a single nonzero 
entry b_n-ma_n-m in position (2n-m,2n-m), and then applies 
Lemma <ref> with p=2n-m-1 and s=n-1). 

Thus the matrix 
S^H:=S(D̃_m,∂D̃_m/∂ a_n-m,a_n-m) 
contains only the following nonzero entries, in the following positions:


    [            Ω _1           (j,j)               ,   Ω _2a_n-1^n-m       (j,j+n-1)               ,    j=1,… ,n-1 ,;                ;           nΩ _1     (ν +n-1,ν )               ,   Ω _2a_n-1^n-m (ν +n-1,ν +n-1)               ,     ν =1,… ,n . ]

One can subtract the (j+n-1)st row from the jth one, j=1, …, 
n-1, to make disappear the terms Ω _2a_n-1^n-m in the first 
n-1 rows; the terms Ω _1 become (1-n)Ω _1. Hence 
S^H=Ω _3a_n-1^n(n-m)≢0. 

Case 2: 1≤ k≤ n-m-1. 
To prove that Δ is not divisible by a_n^n-m-k+1 
we develop it w.r.t. 
its first column:


    Δ := (± b_k^n(1-b_0/b_k)^ka_n^n-m-k)( ([Y]_1,1)+
    (-1)^n+1n ([Y]_n,1)) .

Our aim is to show that for a_n=0 the sum Z:= ([Y]_1,1)+
(-1)^n+1n ([Y]_n,1) is nonzero; this implies a_n^n-m-k+1 not dividing 
Δ. Notice that for a_n=0 the only 
nonzero entries in the second column of Y (i.e. of 
Y|_a_n=0=:Y^0) are Y^0_1,2 and 
Y^0_n,2=(n-1)Y^0_1,2. Thus 


    Z|_a_n=0=(Y^0_1,2+(-1)^n+1(-1)^nnY^0_n,2) (Y^†)=
    (1-n(n-1))Y^0_1,2( Y^†) ,

where the matrix Y^† is obtained from Y^0 
by deleting its first two columns, its first and its nth rows. 


The entry Y^0_1,2 is a not identically equal to 0 polynomial in the 
variables a_j, k≠ j≠ n.
 


Indeed, this is the coefficient of 
a_k^n-1 in R^0:=Res(P,P_*)|_a_n=0. 
The matrix S_*:=S(P,P_*)|_a_n=0 has a 
single nonzero entry in its last column; 
this is (S_*)_2n-m,2n-m=b_n-ma_n-m. Hence R^0=b_n-ma_n-m M, where 
M:=[S_*]_2n-m,2n-m (M is (2n-m-1)× (2n-m-1)). 

For ν =1,… ,n-m one can subtract the (n-m+ν)th row of M 
multiplied by 1/b_k from its 
νth row to make disappear the terms a_k in its first n-m rows. The new 
matrix is denoted by M^1; one has M= M^1. 
The only terms of M^1 containing a_k^n-1 
are now 
obtained by multiplying the entries b_ka_k of the last n-1 rows of M^1. 
To get these terms up to a sign one has to multiply (b_ka_k)^n-1 by 
M^*, where M^* is obtained from M^1 by deleting the rows and 
columns of the entries b_ka_k. The matrix M^* is block-diagonal, its 
left upper block is upper-triangular and its right lower block is 
lower-triangular. The diagonal entries of these blocks (of sizes k× k 
and (n-m-k)× (n-m-k)) equal 1-b_0/b_k and a_n-1. Hence 
Y^0_1,2=± b_n-ma_n-m(1-b_0/b_k)^kb_k^n-1a_n-1^n-m-k≢0.   


There remains to prove that Y^†≢0, see (<ref>). 
The matrix Y^† is obtained as follows. 
Set D^†:=D̃_m|_a_n=0= S_*; 
recall that S_*=b_n-ma_n-m M^1, see the proof of Lemma <ref>. 
Then Y^†=S(D^†,∂ D^†/∂ a_k,a_k). Notice that 
D^† is a degree n-1, not n, polynomial in a_k, therefore 
Y^† is (2n-3)× (2n-3). It suffices to show that for 
a_j=0, j≠ k, n-m, n-1, one has Y^†≢0. This 
results from M^1|_a_j=0,k≠ j≠ n-1 not having multiple roots 
(which we prove below).
 
One can develop n-m-k times M^1 w.r.t. its last column, 
where it has a single 
nonzero entry a_n-1, to obtain 
M^1=± a_n-1^n-m-k M^†; M^† is 
(n+k-1)× (n+k-1), it is obtained from M^1 by 
deleting the last n-m-k columns and the rows with indices 
k+1, …, n-m. The matrix 
M^† satisfies the conditions of Lemma <ref> with 
p=n+k-1 and s=k, the entries r_j from the lemma equal 1-b_0/b_k≠ 0 
(for j=1, …, k) or b_ka_k (for j=k+1, …, n+k-1); 
one has q_j=a_n-1 (1≤ j≤ k) or q_j=b_0 (k+1≤ j≤ n+k-1). 
Hence M^†|_a_j=0,k≠ j≠ n-1=
(1-b_0/b_k)^k(b_ka_k)^n-1± b_0^n-1a_n-1^k. For 
a_n-1≠ 0 it has n-1 distinct roots. Part (3) is proved. 
 


We use sometimes the same notation as in the proof 
of part (3), but with different values of the indices, therefore the proofs 
of the two parts of the proposition should be considered as independent ones. 
For k=n-m+1, … ,n, the polynomial 
D̃_m contains the monomial 
N_k-n+m:=± b_n-m^n-ka_n-m^n-kb_0^ka_k^n-m; it   
does not contain any other monomial of 
the form Ω a_k^n-mD, where D is a product of powers 
of variables a_i with i≠ k, see Proposition <ref>. 













The first column of the (2n-2m-1)× (2n-2m-1)-matrix 
Y:=S(D̃_m,∂D̃_m/∂ a_k,a_k) contains only two 
nonzero entries, namely Y_1,1=± b_n-m^n-ka_n-m^n-kb_0^k  
and Y_n-m,1=± (n-m)b_n-m^n-ka_n-m^n-kb_0^k. Thus 
Y is divisible by a_n-m^n-k. We consider two cases:

Case 1: k=n. We show that Y≢0 if a_n-m=0. We prove this for 
a_j=0, n-m-1≠ j≠ n. In this case the nonzero entries of the matrix 
S(P,P_*) and their positions are


    [                 1             (j,j)                 ,           a_n-m-1       (j,j+n-m-1)                 ,;                  ;               a_n           (j,j+n)                 ,      j=1,…, n-m ,;                  ;               b_0       (ν +n-m,ν )                 ,    b_n-m-1a_n-m-1 (ν +n-m,ν +n-m-1)                 ,       ν =1,… ,n . ]

Subtracting the (j+n-m)th row multiplied by 1/b_n-m-1 from the jth one 
for j=1, …, n-m, one makes disappear the terms a_n-m-1 in the 
first n-m rows. The only nonzero entry in the last column is now a_n in 
position (n-m,2n-m), so 


    S(P,P_*)=(-1)^na_n [S(P,P_*)]_n-m,2n-m .

The last matrix satisfies the conditions of Lemma <ref> with 
p=2n-m-1, s=n and one finds that its determinant is of the form 
Ω _4a_n-m-1^n+Ω _5a_n^n-m-1. Hence 
S(P,P_*)=(-1)^na_n(Ω _4a_n-m-1^n+Ω _5a_n^n-m-1). 
This means that the matrix 
S(D̃_m,∂D̃_m/∂ a_n,a_n) has only the 
following entries in the following positions:


    [                Ω _5               (j,j)                   ,       Ω _4a_n-m-1^n         (j,j+n-m-1)                   ,;                    ;                            j=1,… ,n-m-1                   ,;                    ;           (n-m)Ω _5       (ν +n-m-1,ν )                   ,       Ω _4a_n-m-1^n (ν +n-m-1,ν +n-m-1)                   ,;                    ;                           ν =1,… ,n-m . ]

One can subtract the (j+n-m-1)st row from the jth one, j=1, 
…, n-m-1, to make disappear the terms Ω _4a_n-m-1^n in the 
first n-m-1 rows. The matrix becomes lower-triangular, with diagonal 
entries equal to (1-n+m)Ω _5 or to Ω _4a_n-m-1^n, so its 
determinant is not identically equal to 0.  

Case 2: n-m+1≤ k≤ n-1. To prove that Y is not 
divisible by a_n-m^n-k+1 
we develop it w.r.t. 
its first column:


    Y = (± b_n-m^n-ka_n-m^n-kb_0^k)( ([Y]_1,1)+
    (-1)^n-m(n-m) ([Y]_n-m,1)) .

Our aim is to show that for a_n-m=0 the sum U:= ([Y]_1,1)+
(-1)^n-m(n-m) ([Y]_n-m,1) is nonzero; this implies 
a_n-m^n-k+1 not dividing 
Y. Notice that for a_n-m=0 the only 
nonzero entries in the second column of 
Y^0:=Y|_a_n-m=0 are Y^0_1,2 and 
Y^0_n-m,2=(n-m-1)Y^0_1,2. Thus 


    [                                   U|_a_n-m=0                                            = (Y^0_1,2+(-1)^n+1(-1)^n-m(n-m)Y^0_n-m,2) Y^†;                                             ;                                                                                         =                (1-(n-m)(n-m-1))Y^0_1,2 Y^† , ]

where the matrix Y^† is obtained from Y^0 
by deleting its first two columns, its first and (n-m)th rows. 


The entry Y^0_1,2 is a not identically equal to 0 polynomial in the 
variables a_j, k≠ j≠ n-m.

 
Indeed, this is the coefficient of 
a_k^n-m-1 in R^0:=Res(P,P_*)|_a_n-m=0. 
The matrix S_*:=S(P,P_*)|_a_n-m=0 has a 
single nonzero entry in its last column; 
this is (S_*)_n-m,2n-m=a_n. Hence R^0=a_n M, where 
M:=[S_*]_n-m,2n-m (M is (2n-m-1)× (2n-m-1)). 
 
The only terms of M containing a_k^n-m-1 
are  
obtained by multiplying the entries a_k of the first n-m-1 rows of M. 
To obtain these terms up to a sign one has to multiply a_k^n-m-1 by 
M^*, where M^* is obtained from M by deleting the rows and 
columns of the entries a_k. The matrix M^* is block-diagonal, its 
left upper block is upper-triangular and its right lower block is 
lower-triangular. The diagonal entries of these blocks (of sizes k× k 
and (n-m-k)× (n-m-k)) equal b_0 and a_n-m-1. Hence 
Y^0_1,2=± a_nb_0^ka_n-m-1^n-m-k≢0.  




There remains to prove that Y^†≢0, see (<ref>). 
The matrix Y^† is obtained as follows. 
Set D^†:=D̃_m|_a_n-m=0= S_*; 
recall that S_*=a_n M (see the proof of Lemma <ref>). 
Then Y^†=S(D^†,∂ D^†/∂ a_k,a_k). Notice that 
D^† is a degree n-m-1, not n-m, polynomial in a_k, therefore 
Y^† is (2n-2m-3)× (2n-2m-3). It suffices to show that for 
a_j=0, k≠ j≠ n-m-1, one has Y^†≢0. This 
results from M|_a_j=0,k≠ j≠ n-m-1 not having multiple roots 
(which we prove below).

For a_j=0, k≠ j≠ n-m-1, one can develop n-k times M
w.r.t. its last column in which there is a single nonzero entry 
b_n-m-1a_n-m-1 
(on the diagonal). Hence M=(b_n-m-1a_n-m-1)^n-k M^†, where 
M^† is (n-m+k-1)× (n-m+k-1); it is 
obtained from M by deleting the last n-k rows and columns. 
The matrix M^† satisfies the conditions of Lemma <ref> with 
p=n-m+k-1, s=k, r_j=1, q_j=a_k (j=1, …, n-m-1) or 
r_j=a_n-m-1, q_j=b_0 (j=n-m, …, n-m+k-1). Hence 
M^†=a_n-m-1^k± b_0^ka_k^n-m-1. For a_n-m-1≠ 0 it has 
n-m-1 distinct roots. 









§ SOME PROPERTIES OF THE SETS Θ AND M̃





Suppose that all roots of P_*(.,a^0) (a^0∈𝒜) 
are simple and nonzero    
and that P(.,a^0) and P_*(.,a^0) have exactly one 
root in common. Then for any j=n-m+1, …, n, 
in a neighbourhood of a^0∈𝒜 the set  
{D̃_m=0} is locally the graph of a smooth analytic 
function in the variables a^j. If in addition all roots of P_m,k(.,a^0) 
are simple and nonzero (1≤ k≤ n-m), 
then in a neighbourhood of a^0∈𝒜 the set  
{D̃_m=0} is locally the graph of a smooth analytic 
function in the variables a^k.



Denote by [a]_n-m 

the first n-m coordinates of a∈𝒜. 

Any simple root of P_* is locally (in a 
neighbourhood of [a^0]_n-m) the value of a smooth analytic function 
λ in the variables [a]_n-m. As λ ([a^0]_n-m)≠ 0, 
the condition P(λ ,a)/λ ^j=0, j<m, 
allows to express a_n-j locally (for a_i close to a_i^0, i≠ j) 
as a smooth analytic function 
in the variables a^n-j. Suppose that all roots of P_m,k(.,a^0) 
are simple and nonzero. Then any of these roots is a smooth analytic function 
in the variables a^k. This refers also to μ, the root in common of 
P and P_* which is also a root of P_m,k. Hence one can express 
a_k as a function in a^k from the condition P(μ ,a)/μ ^n-k=0.




At a point of the Maxwell stratum the hypersurface {D̃_m=0} is 
locally the transversal intersection of two smooth analytic hypersurfaces along 
a smooth analytic subvariety of codimension 2.



Suppose first that the roots in common of P and P_* are 0 and 1. 
The two conditions P_*(0)=P_*(1)=0 define a codimension 2 
linear subspace 𝒮 in the space 𝒜 
of the variables a. Adding to them the two conditions 
P(0)=P(1)=0 means defining a codimension 2 linear subspace 
𝒯⊂𝒮; hence 𝒯 is a 
codimension 4 linear subspace of 𝒜. The two linear subspaces 
{ P(0)=0} and { P(1)=0} and their intersections with 
{ P_*(0)=P_*(1)=0} intersect transversally (along respectively 
{ P(0)=P(1)=0} and 𝒯). 

By means of a linear change τ : x↦α x+β, 
α∈ℂ^*, β∈ℂ, one can transform any 
pair of distinct complex numbers into the pair (0,1). Hence at a point of 
𝒯 the Maxwell stratum is locally the direct product of 
𝒯 and the two-dimensional orbit of the group of 
linear diffeomorphisms  
induced in the space 𝒜 by the group of linear changes τ. This 
proves the statement. 



(1) At a point of the set Θ (see Definition <ref>) the set 
{D̃_m=0} is not representable as the graph in the space 
𝒜 of a smooth 
analytic function in the variables a^j, for any j=n-m+1, …, n.

(2) At a point of the set Θ this set is a smooth analytic 
variety of dimension n-2 in the space of variables a.


 
Suppose that for some a=a_0∈𝒜 one has 
P_*(x_0,a_0)=P_*'(x_0,a_0)=0. Suppose first that x_0≠ 0.  
Consider the equation 


    P_*(x,a_0)=ε ,     where    ε∈ (ℂ,0) .
 
Its left-hand side equals 
P_*”(x_0,a_0)(x-x_0)^2/2+o((x-x_0)^2) (with P_*”(a_0,x_0)≠ 0). 
Thus locally (for x close to 
x_0) one has 


    x-x_0=(2/P_*”(x_0,a_0))^1/2ε ^1/2+
    o(ε ^1/2) .
 
In a neighbourhood of a_0∈𝒜 one can introduce new coordinates 
two of which are x_0 and ε. Indeed, one can write








    [                                (n-m-1)!P_*'/n!                                              =         (x-x_0)(x^n-m-2+g_1x^n-m-3+⋯ +g_n-m-2);                                               ;                                                                                             = x^n-m-1+b_1^*a_1x^n-m-2+
    ⋯ +b_n-m-1^*a_n-m-1 , ]

where b_j^*=(n-j)!(n-m-1)!/(n-m-j-1)!n!. 
Hence 


    [           b_1^*a_1                  =            g_1-x_0                  ,           b_2^*a_2                  =         g_2-x_0g_1              , … ,;                   ;   b_n-m-2^*a_n-m-2                  = g_n-m-2-x_0g_n-m-3                  ,   b_n-m-1^*a_n-m-1                  =        -x_0g_n-m-2                  . ]
 
The Jacobian 
matrix ∂ (a_1, …, a_n-m-1)/ ∂ (x_0, g_1, …, 
g_n-m-2) is, up to multiplication of the columns by nonzero constants 
followed by transposition, the Sylvester matrix of the polynomials 
x-x_0 and x^n-m-2+g_1x^n-m-3+⋯ +g_n-m-2. Its determinant is nonzero 
because x_0 is not a root of the second of these polynomials. 

Thus in the space of the variables (a_1, …, a_n-m-1) one can choose 
as coordinates (x_0, g_1, …, g_n-m-2). The polynomial P_* is 
a primitive of P_*' and (-ε ) can be considered as the 
constant of integration, see (<ref>), therefore 
(x_0, g_1, …, g_n-m-2, ε ) can be chosen as 
coordinates in the space of the variables (a_1, …, a_n-m). Adding 
to them (a_n-m+1, …, a_n), one obtains local coordinates 
in the space 𝒜. 

Hence the double root μ of P_* is not an analytic, 
but a multivalued function of the local coordinates in 𝒜, see 
(<ref>). Consider the condition P(μ ,a)/μ ^n-j=0. One can 
express from it a_j (n-m+1≤ j≤ n) 
as a linear combination of the variables a^j with coefficients depending 
on μ. This expression is of the form A+ε ^1/2B, where A 
and B (B≢0) 
depend analytically on the local coordinates in 𝒜. 
This proves the statement for x_0≠ 0. For x_0=0 the 
statement also holds true – if for x_0=0 the set {D̃_m=0} 
is locally the graph of a holomorphic function in the variables a^j, then 
this must be the case for nearby values of x_0 as well which is false.  
Such values exist – the change x↦ x+δ, δ∈ℂ, 
shifts simultaneously by -δ all roots of P 
(hence of all its nonconstant derivatives as well).


 
Denote by ξ the root of P_*' which is 
also a root of P_* and of P. Then ξ is a smooth analytic function 
in the variables a^†:=(a_1, …, a_n-m-1). The condition 
P_*(ξ ,a)=0 allows to express a_n-m as a smooth analytic function 
α in the variables a^†. Set a^*:=a|_a_n-m=α (a^†). 
One can express a_n as a 
smooth analytic function in the variables a_j, n-m≠ j≠ n, from the 
condition P(ξ ,a^*)=0. Thus locally Θ is the graph of a smooth 
analytic vector-function in the variables a_j, n-m≠ j≠ n, 
with two components.  



For 2≤ m≤ n-2 the polynomials B_m,k and C_m,k 
defined in Theorem <ref> are irreducible.



Irreducibility of the factor B_m,k is proved by analogy with 
Proposition <ref>. (For n-m+1≤ k≤ n the analogy is complete 
because after the dilatations a_j↦ a_j/b_j, j=1, …, n-m, 
the polynomial P_* becomes b_0P^*, where P^* is the polynomial 
P defined for n-m instead of n. For 1≤ k≤ n-m the coefficients 
of the polynomial P_m,k are not a_j (we set a_0=1), 
but (b_k-b_j)a_j, and one can perform similar dilatations. 
Only the variable a_k is absent; this, however, is not an obstacle 
to the proof of irreducibility. The details are left for the reader.)

Irreducibility of the factors C_m,k can be proved like this. Denote by 
ξ and η two of the roots of P_*. They are multivalued functions 
of the coefficients a_1, …, a_n-m. The system of two equations 
P(ξ ,a)=P(η ,a)=0 allows to express for ξ≠η the 
coefficients a_n and a_n-1 as functions of a_1, …, a_n-2. 
These multivalued functions are defined over a Zariski dense open 
subset of the space of variables (a_1, …, a_n-2) from 
which irreducibility of the set M̃ follows. Hence its projections 
in the hyperplanes 𝒜^k are also irreducible. 



In the case m=1 one cannot prove in the same way as above that the 
polynomials C_1,k are irreducible because the coefficient a_n-1 is 
in fact a_n-m.




Proposition <ref>, Lemma <ref>, 
Statements <ref>, 
<ref> and <ref> 
allow to conclude that D̃_m,k is of the form 
A_m,kB_m,k^s_m,kC_m,k^r_m,k, 
where s_m,k, r_m,k∈ℕ. Indeed, the form of the factor 
A_m,k is justified by Proposition <ref>. 
It follows from Lemma <ref> and its proof 
that for A_m,kB_m,kC_m,k≠ 0 the polynomials D̃_m and 
∂D̃_m/∂ a_k, when considered as polynomials in a_k, 
have no root in common. Hence a priori D̃_m,k is of the form 
A_m,kB_m,k^s_m,kC_m,k^r_m,k, with 
s_m,k, r_m,k∈ℕ∪ 0 (implicitly we use the irreducibility 
of B_m,k and C_m,k here). 
Statements <ref> and <ref> 
imply that one cannot have s_m,k=0 or r_m,k=0. To prove formula 
(<ref>) now means to prove that s_m,k=1, r_m,k=2. This is 
performed in the next sections.
 




§ THE CASE M=N-2




For m=n-2, n≥ 4, one has s_m,k=1 and r_m,k=2.



For 3≤ k≤ n the polynomial D̃_n-2 is a degree 2 
polynomial in a_k, see Proposition <ref>, so one can set 
D̃_n-2:=Ua_k^2+Va_k+W and 
∂D̃_n-2/∂ a_k=2Ua_k+V, 
where U, V, W∈ℂ[a^k], U≢0. Hence 
S(D̃_n-2,∂D̃_n-2/∂ a_k,a_k)=
( [  U  V  W; 2U  V  0;  0 2U  V ]) and 
D̃_n-2,k=U(4UW-V^2). The second factor is up to a sign the 
discriminant of the quadratic polynomial (in the variable a_k) 
Ua_k^2+Va_k+W. Up to a sign, U is 
the determinant of the matrix S^L obtained from S(P,P_*) by deleting its 
first two rows and the columns, where its entries a_k are situated. 
Hence U=ω a_2^n-k, ω∈ℂ^*. Indeed, S^L is 
block-diagonal, with diagonal blocks of sizes k× k (upper left) and 
(n-k)× (n-k) (lower right). They are respectively upper- and 
lower-triangular, with diagonal entries equal to b_0 and b_2a_2.


For a_2=0 the factor 4UW-V^2 reduces to -V^2∈ℂ[a]. From the 
following lemma we deduce (after its proof) that the factor C_n-2,k 
must be squared. 


The polynomial -V^2 is a 
quadratic polynomial in the variables a_i, i=3, …, n, with the square 
of at least one of them present in -V^2. For k<n (resp. k=n) 
it contains the monomial a_n^2(b_0)^2k(b_1a_1)^2(n-k) 
(resp. a_n-1^2(b_0)^2(n-1)(b_1a_1)^2).
 


Indeed, if k<n, then 
set V^*:=V|_a_j=0,j≠ 1, k, n and S^*:=S(P,P_*)|_a_j=0,j≠ 1, k, n. 
There are two entries a_k (resp. a_1 and a_n) 
in S^*, in positions (1,k+1) and 
(2,k+2) (resp. (1,2), (2,3), and (1,n+1), (2,n+2)). The other 
nonzero entries of S^* are b_0 (resp. b_1a_1) 
in positions (ν +2,ν ) (resp. (ν +2,ν +1)), ν =1, …, 
n. Thus 


    V^*= V^**+ V^***  ,   where  V^**=[S^*]_1,k+1|_a_k=0  ,  
    V^***=[S^*]_2,k+2|_a_k=0 .

The matrices V^** and V^*** are (n+1)× (n+1). Hence 


    V^**=(-1)^na_n [V^**]_1,n+1  ,   V^***=0

(because all entries in the last column of V^*** equal 0). The matrix 
[V^**]_1,n+1 is block-diagonal, with diagonal blocks of sizes 
k× k (left upper, it is upper-triangular) and (n-k)× (n-k) 
(right lower, it is lower-triangular). Their 
diagonal entries equal respectively b_0 and b_1a_1. Thus 


    V^*= V^**=(-1)^na_n(b_0)^k(b_1a_1)^n-k .

Hence for k<n the term -V^2 contains the monomial 
a_n^2(b_0)^2k(b_1a_1)^2(n-k). 

For k=n we set a_j=0, j≠ 1, n-1, n,  
S^†:=S(P,P_*)|_a_j=0,j≠ 1, n-1, n and  
V^†:=V|_a_j=0,j≠ 1, n-1, n. Hence 


    V^†= V^††+ V^†††  ,   where  
    V^††=[S^†]_1,n+1|_a_n=0  ,  
    V^†††=[S^†]_2,n+2|_a_n=0 .

One has V^††=0 (all entries in the last column are 0) and 
V^††† has an entry a_n-1 in position (1,n); no other 
entry of V^††† depends on a_n-1. Hence 
V^††† contains the monomial 
(-1)^n+1a_n-1 [V^†††]_1,n. The matrix 
[V^†††]_1,n is block-diagonal, with diagonal blocks of 
sizes (n-1)× (n-1) (upper left, it is upper-triangular, with 
diagonal entries equal to b_0) and 
1× 1 (lower right, it equals b_1a_1). Hence -V^2 contains the 
monomial a_n-1^2(b_0)^2(n-1)(b_1a_1)^2.


The factor C_n-2,k is a linear function in the variables 
a_3, …, a_n, with coefficients depending on a_1 and a_2. Indeed, 
set P_*:=b_0(x-α )(x-β ), 0≠α≠β≠ 0. 
One can choose (α ,β ) as 
coordinates in the space (a_1,a_2). The polynomial P is obtained from 
P_* by rescaling of its coefficients followed by (n-2)-fold integration 
with constants of integration 
of the form η _sa_s, η _s∈ℚ^*, s=3, …, n. 
Consider the two conditions P(α ,a)/α ^n-k=0 and 
P(β ,a)/β ^n-k=0. Each of them is a linear form in the variables 
a_3, …, a_n, with coefficients depending on a_1 and a_2; the 
one of a_k equals 1. 
The projection of the Maxwell stratum in the space of the variables a^k is 
given by the condition 


    β ^n-kP(α ,a)-α ^n-kP(β ,a)=0 .
  
Its left-hand side is a linear form in the variables a_3, …, a_k-1, 
a_k+1, …, a_n, with coefficients depending on α and β. 
The presence of the monomial a_n^2(b_0)^2k(b_1a_1)^2(n-k) or 
a_n-1^2(b_0)^2(n-1)(b_1a_1)^2 in D̃_n-2,k (see Lemma <ref>) 
implies that the factor C_n-2,k must be squared. 

There remains to prove that s_n-2,k=1, see Remark <ref>. 
The left-hand side of equation (<ref>) is divisible by α - β. 
Represent this expression in the form (α - β )Q(α ,β ,a). 
The polynomial Q depends in fact on α +β =-b_1a_1/b_0, 
αβ =b_2a_2/b_0 and a, 
hence this is a polynomial in a (denoted by K(a)). 

Clearly K depends 
linearly on the variables a_3, …, a_n. On the other hand 
K is quasi-homogeneous. Hence K is irreducible. Indeed, should K be 
the product of two factors, then one of the two (denoted by Z) 
should not depend on any 
of the variables a_3, …, a_n, i.e. Z should be a polynomial in 
a_1 and a_2. 

This polynomial should divide the coefficients of 
all variables a_3, …, a_n in K. But for 3≤ s≤ n 
the coefficient of a_s in 
K equals (see (<ref>)) 
c_s:=(β ^n-kα ^n-s-α ^n-kβ ^n-s)/(α - β ). 
Hence Z divides c_s-β c_s-1=α ^n-s-1β ^n-k 
for all s≠ k, and by symmetry Z divides α ^n-kβ ^n-s-1 
for all s≠ k. Hence Z=1 and the polynomial C_n-2,k equals 
(β ^n-kP(α ,a)-α ^n-kP(β ,a))/(α -β ). Its 
quasi-homogeneous weight (QHW) is 2n-k-1 (notation: QHW(C_n-2,k)=2n-k-1). 
Indeed, one has to consider QHW(α ) and QHW(β ) to be equal to 
1 because α and β are roots of 
P_* and their QHW is the same as the one of the 
variable x. 

Obviously QHW(D̃_n-2,k)=2QHW(U)+QHW(W) because 
D̃_n-2,k=U(4UW-V^2) and D̃_n-2,k is quasi-homogeneous. 
As U=ω a_2^n-k, one has QHW(U)=2(n-k). The polynomial 
D̃_n-2,k contains a monomial ω̃a_2^n, 
ω̃≠ 0 (see Proposition <ref>). This monomial 
is contained also in W=D̃_n-2|_a_k=0 hence 
QHW(D̃_n-2)=QHW(W)=2n. Thus 


    QHW(D̃_n-2,k)=2 QHW(U)+ QHW(D̃_n-2)
    =6n-4k .

On the other hand one knows already that a priori 
D̃_n-2,k=A_n-2,kB_n-2,k^s_n-2,kC_n-2,k^2, s_n-2,k∈ℕ, 
A_n-2,k=a_2^n-k. Hence 


    [                        s_n-2,k QHW(B_n-2,k)                                           = QHW(D̃_n-2,k)- QHW(A_n-2,k)-
    2 QHW(C_n-2,k);                                                                                       =                      6n-4k-2(n-k)-2(2n-k-1)                                           =                                         2 , ]

and as B_n-2,k=b_1^2a_1^2-4b_0b_2a_2, one has QHW(B_n-2,k)=2, so s_n-2,k=1.



 
In order to deal with the cases k=1 and k=2 we need to know the 
degrees and 
quasi-homogeneous weights of certain polynomials in the variables a:


(1) _a_1D̃_n-2=n, _a_2D̃_n-2=n; 

(2) QHW(D̃_n-2)=2n, 
QHW(∂D̃_n-2/∂ a_1)=2n-1, 
QHW(∂D̃_n-2/∂ a_2)=2n-2; 

(3) QHW(D̃_n-2,1)=n(3n-2); 

(4) QHW(D̃_n-2,2)=2n(n-1);

(5) QHW(B_n-2,1)=n(n-1), QHW(B_n-2,2)=n(n-1); 

(6) QHW(C_n-2,1)=n(n-1), QHW(C_n-2,2)=n(n-1)/2.


For k=1 or 2 one has to find positive integers u and v such that 


    QHW(D̃_n-2,k)=(2-k)n+u QHW(B_n-2,k)+v QHW(C_n-2,k) ,

because A_n-2,k=a_n^2-k. For k=2 parts (4), (5) and (6) of the lemma 
imply that 
u=1, v=2 
is the only possible choice. For k=1 there remain two possibilities – 
(u,v)=(1,2) or (u,v)=(2,1) – so we need another lemma as well:



For a_j=0, j≠ 1, n-1, n, the polynomials D̃_n-2, 
D̃_n-2,1, B_n-2,1 and C_n-2,1 are of the form respectively 
(with Δ _i≠ 0)

    [                                 D̃_n-2                                      = Δ _1a_na_1^n+Δ _2a_na_n-1a_1+Δ _3a_n^2                                      ,                               D̃_n-2,1                                      =       Δ _4a_n^2n-1a_n-1^n+Δ _5a_n^3n-2                                      ,;                                       ;                                B_n-2,1                                      =                Δ _6a_n^n-1+Δ _7a_n-1^n                                    and                                C_n-2,1                                      =                            Δ _8a_n^n-1                                      . ]



The lemma implies that it is possible to have (u,v)=(1,2), but not 
(u,v)=(2,1). Indeed, otherwise the product 
D̃_n-2,1=A_n-2,1B_n-2,1^2C_n-2,1, with 
A_n-2,1=a_n, should contain three different monomials
whereas it contains only two.




Parts (1) and (2) follow directly from Proposition <ref>. To prove 
parts (3) and (4) one has to observe that as the polynomial D̃_n-2 
contains a monomial c^*a_n^2, c^*≠ 0, 
the (2n-1)× (2n-1)-Sylvester matrices 
S^*_k:=S(D̃_n-2,∂D̃_n-2/∂ a_k,a_k), 
k=1 or 2, 
contain this monomial 
in positions (j,j+n), j=1, …, n-1 and only there. The matrix 
S^*_1 (resp. S^*_2) has entries c^†a_n, c^†≠ 0
(resp. c^**≠ 0) 
in positions 
(ν +n-1,ν ), ν =1, …, n. Hence 
D̃_n-2,k contains a 
monomial ± (c^†a_n)^n(c^*a_n^2)^n-1 for k=1 and 
± (c^**)^n(c^*a_n^2)^n-1 for k=2 whose quasi-homogeneous weight is 
respectively n(3n-2) and 2n(n-1). 

To prove part (5) recall that the 
(2n-1)× (2n-1)-Sylvester matrix 
S^0:=S(P_n-2,k,P_n-2,k'), k=1 or 2, 
has entries of the form c^**a_n, c^**≠ 0, in positions 
(j,j+n), j=1, …, n-1 and only there, and constant nonzero terms 
in positions (ν +n-1,ν ), ν =1, …, n. Thus B_n-2,k 
contains a monomial ± c^***(a_n)^n-1, c^***≠ 0 and  
QHW(B_n-2,k)=n(n-1). 





 
For the proof of part (6) we need to recall that the factors C_n-2,k are 
related to polynomials P divisible by P_*. When one performs this 
Euclidean division one obtains a rest of the form 
U^†(a)x+V^†(a), where 
U^†,V^†∈ℂ[a], QHW(U^†)=n-1, 
QHW(V^†)=n, U^† (resp. V^†) 
contains monomials 
ω _1a_1^n-1 and ω _2a_n-1 (resp. ω _3a_1^n-2a_2 and 
ω _4a_n), 
ω _i≠ 0. (To see that the monomials ω _1a_1^n-1 and 
ω _3a_1^n-2a_2 are present one has to recall that at each step of the 
Euclidean division one replaces a term Lx^s, L∈ℂ[a], 
by the sum 
-L(b_1/b_0)a_1x^s-1-L(b_2/b_0)a_2x^s-2.) 

To obtain the factor C_n-2,1 one has to eliminate a_1 from the system 
of equations U^†(a)=V^†(a)=0, i.e. one has to find the subset 
in the space of variables a^1 for which U^† and V^† 
have a common zero when 
considered as polynomials in a_1. The (2n-3)× (2n-3)-Sylvester 
matrix S(U^†,V^†,a_1) contains terms ω _2a_n-1 
in positions 
(j,j+n-1), j=1, …, n-2, and terms ω _3a_2 in positions 
(ν +n-2,ν ), ν =1, …, n-1. Hence C_n-2,1 contains a 
monomial ± (ω _2a_n-1)^n-2(ω _3a_2)^n-1, of 
quasi-homogeneous weight n(n-1). 

The proof of the second statement of part (6) is performed separately 
for the cases of even and odd n. If n is even, then U^† 
(resp. V^†) 
contains monomials Ω _1a_1a_2^n/2-1 and Ω _2a_n-1 
(resp. Ω _3a_2^n/2 and Ω _4a_n), Ω _i≠ 0. The 
(n-1)× (n-1)-Sylvester matrix  S(U^†,V^†,a_2) 
contains terms 
Ω _4a_n in positions 
(j,j+n/2), j=1, …, n/2-1, and Ω _1a_1 in positions 
(ν +n/2-1,ν ), ν =1, …, n/2. Hence C_n-2,2 contains a 
monomial ± (Ω _4a_n)^n/2-1(Ω _1a_1)^n/2, of 
quasi-homogeneous weight n(n-1)/2. 

When n is odd, then U^† (resp. V^†) 
contains monomials Ω̃_1a_2^(n-1)/2 and Ω̃_2a_n-1 
(resp. Ω̃_3a_1a_2^(n-1)/2 and Ω̃_4a_n), 
Ω̃_i≠ 0. The 
(n-1)× (n-1)-Sylvester matrix  
S(U^†,V^†,a_2) contains terms 
Ω̃_2a_n-1 in positions 
(j,j+(n-1)/2), j=1, …, (n-1)/2, and 
Ω̃_3a_1 in positions 
(ν +(n-1)/2,ν ), ν =1, …, (n-1)/2. Thus 
C_n-2,2 contains a monomial 
± (Ω̃_2a_n-1Ω̃_3a_1)^(n-1)/2, of 
quasi-homogeneous weight n(n-1)/2.






One can develop S(P,P_*) w.r.t. the last column in which there 
is a single nonzero entry (a_n, in position (2,n+2)). Hence 
D̃_n-2=(-1)^na_n S^♯, where 
S^♯:=[S(P,P_*)]_2,n+2. The last column of S^♯ contains only 
two nonzero entries (a_n in position (1,n+1) and b_1a_1 
in position (n+1,n+1)), therefore 


    S^♯=(-1)^na_n S^♯ 1+b_1a_1 S^♯ 2  ,   where  S^♯ 1:=[S^♯]_1,n+1  ,  
    S^♯ 2:=[S^♯]_n+1,n+1 .

The matrix S^♯ 1 is upper-triangular, with diagonal 
entries equal to b_0, so S^♯ 1=b_0^n, while S^♯ 2 contains 
only two nonzero entries in its last column (a_n-1 in position (1,n) and 
b_1a_1 in position (n,n)). Hence 


    S^♯ 2=(-1)^n+1a_n-1 S^♯ 3+b_1a_1 S^♯ 4  ,   where  S^♯ 3:=[S^♯ 2]_1,n  ,  
    S^♯ 4:=[S^♯ 2]_n,n .

The matrix S^♯ 3 is upper-triangular, with diagonal entries equal to 
b_0, so S^♯ 3=b_0^n-1. The matrix S^♯ 4 becomes 
lower-triangular after subtracting its second row multiplied by 1/b_1 from 
the first one, 
with diagonal entries 1-b_0/b_1, b_1a_1, …, b_1a_1, from which 
the form of D̃_n-2 follows. 

Hence the (2n-1)× (2n-1)-Sylvester matrix 
S(D̃_n-2,∂D̃_n-2/∂ a_1,a_1) has only the 
following nonzero entries, in the following positions:


    [            Δ _1a_n            (j,j) ,       Δ _2a_na_n-1        (j,j+n-1) ,          Δ _3a_n^2          (j,j+n) ,;                   ;                           j=1,… ,n-1,;                   ;           nΔ _1a_n      (ν +n-1,ν ) ,       Δ _2a_na_n-1 (ν +n-1,ν 
    +n-1) ,        ν =1,… ,n . ]

One can subtract the (j+n-1)st row from the jth one (j=1,… ,n-1) 
to make disappear the terms Δ _2a_na_n-1 in positions 
(j,j+n-1). This does not change the determinant; the entries 
Δ _1a_n in positions (j,j) become (1-n)Δ _1a_n. The form of 
D̃_n-2,1 follows now from Lemma <ref>.  

For a_j=0, j≠ 1, n-1, n, the polynomial P_n-2,1 is of the form 
α _1x^n+α _2a_n-1x+α _3a_n, α _i≠ 0, so 
the (2n-1)× (2n-1)-Sylvester matrix S(P_n-2,1,P_n-2,1') 
has nonzero entries only


    [              α _1        at (j,j) ,         α _2a_n-1    at (j,j+n-1) ,           α _3a_n      at (j,j+n) ,       j=1,… ,n-1,;                  ;             nα _1      at (ν ,ν ) ,         α _2a_n-1 at (ν ,ν 
    +n-1) ,       ν =1,… ,n . ]

By analogy with the reasoning about D̃_n-2,1 one finds that 
B_n-2,1=Δ _6a_n^n-1+Δ _7a_n-1^n. 

To justify the form of C_n-2,1 it suffices to observe that for 
a_j=0, j≠ 1, n-1, n, one has (see the definition of 
U^† and V^† in the proof of Lemma <ref>) 
U^†=α _4a_n-1+α _5a_1^n-1, 
V^†=α _6a_n, α _i≠ 0, so _a_1U^†=n-1 and   
_a_1V^†=0. When eliminating a_1 from the system of 
equalities U^†=V^†=0 one obtains Res(U^†,V^†,a_1)=0, 
i.e. (α _6a_n)^n-1=0. 




§ THE PROOF OF S_M,1=1


In the present section we prove the following 


With the notation of Remark <ref> one has s_m,1=1.



The proof of the proposition makes use of the following lemma:


Set a_j=0 for j≠ 1, ℓ and n, where n-m+1≤ℓ≤ n-1. 
Then S(P,P_*) is of the form 
Ω _1a_n^n-m-1a_1^n+Ω _2a_n^n-m-1a_ℓa_1^n-ℓ+Ω _3a_n^n-m.


 
Lemma <ref> with ℓ =n-1 implies that the matrix 
S(D̃_m,∂D̃_m/∂ a_1, a_1) has only the following 
nonzero entries, in the following positions:


    [      Ω _1a_n^n-m-1              (j,j)                  , Ω _2a_n^n-m-1a_n-1       (j,j+n-1)  ,;                   ;        Ω _3a_n^n-m            (j,j+n)                  ,       j=1,… ,n-1 ,;                   ;     nΩ _1a_n^n-m-1        (ν +n-1,ν )                  , Ω _2a_n^n-m-1a_n-1 (ν +n-1,ν +n-1)  ,;                   ;                           ν =1,… ,n . ]

Subtract for j=1, …, n-1 its (j+n-1)st row from the jth one. 
This preserves its determinant and leaves only the following nonzero entries, 
in the following positions:


    [ (1-n)Ω _1a_n^n-m-1              (j,j)                  ,        Ω _3a_n^n-m            (j,j+n)                  ,;                   ;                            j=1,… ,n-1                  ,;                   ;     nΩ _1a_n^n-m-1        (ν +n-1,ν )                  , Ω _2a_n^n-m-1a_n-1    (ν +n-1,ν +n-1)                  ,;                   ;                           ν =1,… ,n .                    ]
 
The new matrix satisfies the conditions of Lemma <ref> with 
p=2n-1, s=n. Hence its determinant is of the form 


    a_n^(n-m-1)(2n-1)(Ω _4a_n-1^n+Ω _5a_n^n-1)  ,

where Ω _4=((1-n)Ω _1)^n-1Ω _2^n and 
Ω _5=±Ω _3^n-1Ω _1^n. The polynomial 
Res(P_m,1,P_m,1') contains monomials α a_n^n-1 and β a_n-1^n, 
α≠ 0≠β; this can be proved by complete analogy with 
the analogous statement of Proposition <ref> with m=1 and we 
leave the proof for the reader. Hence the polynomial (<ref>) is not 
divisible by a power of Res(P_m,1,P_m,1') higher than 1, because in this 
case it would contain at least three different monomials in a_n and 
a_n-1. Thus s_m,1=1.    



The matrix S(P,P_*) has only the following nonzero entries, 
in the following positions:


    [             1         (j,j)             ,           a_1       (j,j+1)             ,           a_ℓ   (j,j+ℓ )  ,;              ;           a_n       (j,j+n)             ,                j=1,… ,n-m ,;              ;           b_0   (ν +n-m,ν )             ,        b_1a_1 (ν +n-m,ν +1)             ,   ν =1,… ,n . ]

One can develop the determinant n-m-1 times w.r.t. the last column in which 
each time there will be a single nonzero entry a_n. Thus 
S(P,P_*)=± a_n^n-m-1 S^, where the first row of 
S^ contains the entries 1, a_1, a_ℓ and a_n in positions 
respectively (1,1), (1,2), (1,ℓ +1) and (1,n+1); its second row 
is of the form (b_0, b_1a_1, 0, …, 0) and the next rows are 
the consecutive shifts of this one by one position to the right. Developing 
of S^ w.r.t. the last column yields 


    S^=(-1)^na_n [S^]_1,n+1+b_1a_1
     [S^]_n+1,n+1 .

The matrix [S^]_1,n+1 is upper-triangular, with diagonal entries 
equal to b_0 (hence [S^]_1,n+1=b_0^n). The determinant of the 
matrix S^:=[S^]_n+1,n+1 can be developed 
n-ℓ -1 times 
w.r.t. its last column, where each time 
it has a single nonzero entry b_1a_1 in its right lower corner:


    S^=(b_1a_1)^n-ℓ -1 S^*† ,

where S^*† is (ℓ +1)× (ℓ +1); it is obtained by 
deleting the last n-ℓ -1 rows and columns of S^. The 
determinant S^*† can be developed w.r.t. its last column: 


    S^*†=(-1)^ℓa_ℓ [S^*†]_1,ℓ +1+b_1a_1
     [S^*†]_ℓ +1,ℓ +1 .

The matrix [S^*†]_1,ℓ +1 (resp. [S^*†]_ℓ +1,ℓ +1) 
is upper-triangular, with diagonal entries equal to b_0, so 
its determinant equals b_0^ℓ (resp. becomes lower-triangular (after 
subtracting its second row multiplied by 1/b_1 from its first row), 
with diagonal 
entries equal to 1-b_0/b_1, b_1a_1, …, b_1a_1, so its determinant 
equals (1-b_0/b_1)(b_1a_1)^ℓ -1). This implies the lemma. 





§ COMPLETION OF THE PROOF OF 
THEOREM <REF>


 
If formula (<ref>) is true for n=n_0, k=k_0, 
then it is true for n=n_0+1, k=k_0+1.



If formula (<ref>) is true for n=n_0, m=n_0-2, k=1, then 
it is true for n=n_0, 2≤ m<n_0-2, k=1.



Recall that we have shown already 
(see Remark <ref>) that for 
each n fixed the polynomials D̃_m,k (2≤ m≤ n-2, 
1≤ k≤ n) are of the form A_m,kB_m,k^s_m,kC_m,k^r_m,k, 
s_m,k, r_m,k∈ℕ. Suppose that for 4≤ n≤ n_0 one has 
s_m,k=1, r_m,k=2. (Using MAPLE one can obtain this result  
for n_0=4.) Set P(a,x):=x^n_0+a_1x^n_0-1+⋯ +a_n_0, a:=
(a_1,… ,a_n_0) and consider the polynomials F:=ux^n_0+1+P and 
F_*:=b_-1ux^n_0-m+1+P_*, u∈ (ℂ,0), 0≠ b_-1≠ b_j for 
0≤ j≤ n_0-m. They are deformations 
respectively of P and P_*. Our reasoning uses the following 


One has 


    [                                                            F                                                            =                  u(x^n_0+1+x^n_0/u+∑ _j=0^n_0-1(a_n-j/u)x^j)                                                            ,;                                                             ;                                                          F_*                                                            = u(b_-1x^n_0-m+1+b_0x^n_0-m/u+
    ∑ _j=0^n_0-1(b_n-ja_n-j/u)x^j)                                                            , ]

so after the change of parameters 
ã_1=1/u, ã_s=a_s-1/u, 
s=2, …, n_0 (which is well-defined for u≠ 0) and the 
shifting by 1 of the indices of the constants b_j, the polynomials F and 
F_* (up to multiplication by 1/u) become P and P_* defined for n_0+1 
instead of n_0.
  


The zero set of Res(F,F_*) for u≠ 0 
is defined by 
an equation of the form D̃_m+uH/d=0, where 
H∈ℂ[u,a] and d≠ 0. 




Consider the 
(2n_0-m+2)× (2n_0-m+2)-Sylvester matrix 
S̃:=S(F,F_*). Permute the rows of S̃ as follows: 
place the (n_0-m+2)nd row in second position while shifting 
the ones with indices 2, …, n_0-m+1 by one position backward. 
This preserves up to a sign the determinant and yields a  
matrix T which we decompose in four blocks the diagonal ones being of size 
2× 2 (upper left, denoted by T^*) and 
(2n_0-m)× (2n_0-m) (lower right, denoted by T^**); the left lower 
block is denoted by T^0 and the right upper by T^1. 
An easy check shows that 


    T^*=( [     u     1; b_-1u   b_0 ])   ,  T^**|_u=0=S(P,P_*)  ,
  
and that the only nonzero entries of the left lower block T^0 are 
u and b_-1u, in positions 
(3,2) and (n_0-m+3,2) respectively.

Divide the first 
column of T by u (we denote the thus obtained matrix by T^†). 
This does not change the zero set of T for 
u≠ 0. For u=0 the matrix T^† is block-upper-triangular, 
with diagonal blocks equal to 
( [    1    1; b_-1  b_0 ]) and S(P,P_*). 
Hence T^†=d S(P,P_*)+uH(u,a), 
d:= T^*|_u=0=b_0-b_-1≠ 0, 
H∈ℂ[u,a]. 
Thus the zero set of Res(F,F_*) for u≠ 0 
sufficiently small is defined by 
the equation D̃_m+uH/d=0.
 

For u≠ 0 (resp. for u=0) the quantity T^† 
is a degree n_0-m+1 (resp. n_0-m) polynomial in 
a_k for k=n_0-m+1, …, n_0, and a degree n_0+1 (resp. 
n_0) polynomial in a_k for k=1, …, n_0-m, see 
Proposition <ref>. Hence for each k=1, …, n_0 
there is one simple root -1/w_k(u,a) of Res(F,F_*) 
that tends to infinity as u→ 0. Thus one can set 
Res(F,F_*)=(1+w_k(u,a)a_k)D̃^*_m, where 
D̃^*_m|_u=0≡D̃_m and 
_a_kD̃^*_m=n_0-m (resp. n_0) 
for k=n_0-m+1, …, n_0 (resp. for k=1, …, n_0-m). 


Set E_m:=Res(F,F_*) and 
D̃^*_m,k:=Res(E_m,∂ E_m/∂ a_k,
a_k). Then for u≠ 0 one has   
D̃^*_m,k=Ω ^♭♭(a_n_0^2(n_0-m-k)D̃_m,k
+uH_m,k(u,a)), where H_m,k∈ℂ[u,a].



One can set u:=a_n_0^2(n_0-m-k)v to obtain the equality 

    D̃^*_m,k=Ω ^♭♭a_n_0^2(n_0-m-k)(D̃_m,k
    +vH_m,k(a_n_0^2(n_0-m-k)v,a)) .
 
Now in a neighbourhood of each a_n_0≠ 0 
fixed the zero set of D̃^*_m,k is defined by the equation 
D̃_m,k+vH_m,k(a_n_0^2(n_0-m-k)v,a)=0, i.e. by deforming the equation 
D̃_m,k=0.


 
Indeed, Proposition <ref> implies that D̃_m 
contains a monomial Ω ^♭a_k^n_0a_n_0^n_0-m-k, 1≤ k≤ n_0-m 
(resp. Ω _♭a_k^n_0-ma_n_0-m^n_0-k, n_0-m+1≤ k≤ n_0) 
and this is the only 
monomial containing a_k^n_0 (resp. a_k^n_0-m). Similarly, E_m contains 
a monomial I:=u^k+1Ω ^♮a_k^n_0+1a_n_0^n_0-m-k, 1≤ k≤ n_0-m 
(resp. J:=u^k+1Ω _♮a_k^n_0-m+1a_n_0-m^n_0-k, 
n_0-m+1≤ k≤ n_0) 
and this is the only monomial containing a_k^n_0+1 (resp. a_k^n_0-m+1). 
(The monomial I is obtained as follows: one subtracts for ν =1, …, n_0-m+1 
the (ν +n_0-m+1)st row multiplied by 1/b_k from the νth one to make disappear 
the terms a_k in the first n_0-m+1 rows. The monomial I is the product of the terms 
b_ka_k in the last n_0+1 rows, the terms (1-1/b_k)u in the first k+1 rows and the terms 
a_n_0 in the next n_0-m-k rows. The monomial J is obtained in a similar way. One has 
to assume that QHW(u)=-1.) 
Knowing that _a_kE_m=n_0+1 (resp. _a_kE_m=n_0-m+1) for u≠ 0 
and that _a_kD̃_m=n_0 (resp. _a_kD̃_m=n_0-m) 
one concludes that 


    [                                                                 E_m                                 =   u^k+1Ω ^♮a_k^n_0+1a_n_0^n_0-m-k                                 +          Ω ^♭a_k^n_0a_n_0^n_0-m-k                                 +                         uE^*(u,a);                                  ;                            (resp.                               E_m                                 = u^k+1Ω _♮a_k^n_0-m+1a_n_0-m^n_0-k                                 +        Ω _♭a_k^n_0-ma_n_0-m^n_0-k                                 +                  uE^**(u,a)   ) , ]

where E^*, E^**∈ℂ[u,a], _a_kE^*≤ n_0, 
_a_kE^**≤ n_0-m. The Sylvester matrix 
S(E_m, ∂ E_m/∂ a_k,a_k) is (2n_0+1)× (2n_0+1) (resp. 
(2n_0-2m+1)× (2n_0-2m+1)). We permute its rows by placing the 
(n_0+1)st (resp. (n_0-m+1)st) row in second position while shifting by 
one position backward the second, third, …, n_0th (resp. (n_0-m)th) 
rows. The new matrix T^♭ can be block-decomposed, with diagonal blocks 
T^uℓ (2× 2, upper left) and T^ℓ r; the other two blocks are 
denoted by T^ur and T^ℓℓ. Hence


    [                                                                                                                                                                                                                                                                                                                                                                            T^uℓ                                                                                                                                                                                       =                     ( [         u^k+1Ω ^♮a_n_0^n_0-m-k    Ω ^♭a_n_0^n_0-m-k+uX^1(u,a);                               ;  (n_0+1)u^k+1Ω ^♮a_n_0^n_0-m-k n_0Ω ^♭a_n_0^n_0-m-k+uX^2(u,a) ])                                                                                                                                                                                       ,;                                                                                                                                                                                        ;                                                                                                                                                                                  (resp.                                                                                                                                                                                    T^uℓ                                                                                                                                                                                       = ( [             u^k+1Ω _♮a_n_0^n_0-m-k        Ω _♭a_n_0^n_0-m-k+uX^3(u,a);                                   ;    (n_0-m+1)u^k+1Ω _♮a_n_0^n_0-m-k (n_0-m)Ω _♭a_n_0^n_0-m-k+uX^4(u,a) ])                                                                                                                                                                                     ) , ]

X^i∈ℂ[u,a]. One has 
T^ℓ r|_u=0=S(D̃_m,∂D̃_m/∂ a_k,a_k). The block 
T^ℓℓ has just two nonzero entries, in its second column, and 
T^ℓℓ|_u=0=0. The first of these entries is in position 
(3,2) and equals u^k+1Ω ^♮a_n_0^n_0-m-k (resp. 
u^k+1Ω _♮a_n_0^n_0-m-k). The second of them is in position 
(n_0+2,2) (resp. (n_0-m+2,2)) and equals 
(n_0+1)u^k+1Ω ^♮a_n_0^n_0-m-k (resp. 
(n_0-m+1)u^k+1Ω _♮a_n_0^n_0-m-k). 

Thus for u=0≠ a_n_0 the zero set of D̃_m,k^* is the one of 
D̃_m,k. For u≠ 0 small enough this set does not change if one 
divides the first column of the matrix T^♭ by u^k+1. We denote the new 
matrix by T^♭ *. Obviously 
T^♭ *=-Ω ^♮Ω ^♭(a_n_0^2(n_0-m-k)D̃_m,k+uH_m,k) (resp. 
T^♭ *=-Ω _♮Ω _♭(a_n_0^2(n_0-m-k)D̃_m,k+uH_m,k)) for a suitably defined polynomial H_m,k which 
proves the lemma.
 


Further to distinguish between the sets Θ and M̃ 
(see Definition <ref>) 
defined for the polynomials P or F we write Θ _P and M̃_P 
or Θ _F and M̃_F. 
Consider a point A∈Θ _P and a germ 
𝒢 of an affine space of dimension 2 which intersects Θ _P 
transversally at A. Hence there exists a compact 
neighbourhood 𝒩 of 
A in the space 𝒜 such that the parallel translates of 
𝒢 which intersect Θ _P at points of 𝒩, 
intersect Θ _P transversally at these points. We assume that the value 
of |a_n_0| remains ≥ρ in 𝒩 for some ρ >0. 
The restrictions of D̃_m,k to each of 
these translates are smooth analytic functions each of which has one 
simple zero at its 
intersection point with Θ _P; this follows from the factor B_m,k 
participating in power 1 in formula (<ref>) for n=n_0. 
Hence for all  
u∈ℂ with 0<|u|≪ρ the restriction of D̃^*_m,k 
to these translates are smooth analytic functions having simple zeros at the 
intersection points of the translates with Θ _P. 

But this means that the power of the factor B_m,k in formula (<ref>) 
applied to the polynomial F is equal to 1 on the intersection of 
Θ _F with some open ball of dimension n_0+1 centered at (0,A) in 
the space of the variables (u,a). 
Hence this power equals 1 on some Zariski open dense subset Θ ^0 
of Θ _F 
(if its complement Θ _F\Θ ^0 is nonempty, then on 
Θ ^0 this power might be >1). 
Thus the equality s_m,k=1 is justified for n=n_0+1, 2≤ k≤ n_0+1 
(because it is the coefficient of x^n_0-k, not of x^n_0+1-k of F,  
that equals a_k). 

Now we adapt the above reasoning to the situation, where instead of a point 
A ∈Θ _P one considers a point A∈M̃_P. Each of the 
translates of 𝒢 intersects M̃_P transversally, at just 
one point. The restriction of D̃_m,k to the translate is a 
smooth analytic function having a double zero, so a priori the restriction of 
D̃^*_m,k to it has either one double or two simple zeros. (Under an  
analytic deformation a double zero either remains such or splits into two 
simple zeros.) 
However two simple zeros is impossible because these zeros would be two 
points of M̃_P whereas the translate contains just one point. Thus 
the power 2 of the factor C_m,k is justified for some 
Zariski open dense subset of M̃_F. Once again, 
this is sufficient to claim that 
formula (<ref>) is valid for n=n_0+1 and for 2≤ k≤ n_0+1. 





Recall that by Remark <ref> we have to show that for n=n_0 
one has s_m,k=1, r_m,k=2. The first of these equalities was proved in 
Section <ref> (see Proposition <ref>), so there remains to prove 
the second one. 

As in the proof of Statement <ref> we set 
P(a,x):=x^n_0+a_1x^n_0-1+⋯ +a_n_0, a:=
(a_1,… ,a_n_0). We define the polynomial P_*:=x^2+b_1a_1x+b_2a_2 to 
correspond to the case m=n_0-2 
(i.e. b_k≠ 0, 1, b_3-k for k=1, 2). 
For m=n_0-2 Theorem <ref> 
is proved in Section <ref>, so we assume that m<n_0-2 and we set 
G:=x^n_0-m-2P_*+u(b_3a_3x^n_0-m-3+⋯ +b_n_0-ma_n_0-m), where 
u∈ (ℂ,0) and for i,j≥ 3, i≠ j, one has 
0≠ b_i≠ b_j≠ 0. 
Denote by G^♯ the (2n_0-m)× (2n_0-m)-matrix 
S(P,G).


One has G^♯|_u=0=a_n_0^n_0-m-2 S(P,P_*)=
a_n_0^n_0-m-2D̃_2. 
Hence G̃:= G^♯=a_n_0^n_0-m-2D̃_2+uH^♯(u,a), 
H^♯∈ℂ[u,a]. 

 

All nonzero entries of the matrix G^♯ in the intersection of 
its last n_0-m-2 columns 
and rows are 0 for u=0. One can develop n_0-m-2 times 
G^♯|_u=0 w.r.t. its last column; 
each time there is a single nonzero entry in it  
which equals a_n_0. The matrix obtained from G^♯|_u=0 by deleting 
its last n_0-m-2 columns and the rows with indices m+2, …, n_0-1 
is precisely S(P,P_*). 


One can observe that G^♯ and G^♯|_u=0 are 
both degree n_0 polynomials in a_1. Assume that a_n_0 belongs to a 
closed disk on which one has |a_n_0|≥ρ ^♭>0. Suppose that 
|u|≪ρ ^♭, so one can consider the quantity 
D̃_2+(u/a_n_0^n_0-m-2)H^♯(u,a)
as a deformation of D̃_2. To this end we set u:=a_n_0^n_0-m-2v, 
v∈ (ℂ,0), see Remark <ref>. 
Now to prove Statement <ref> one has just to repeat 
the reasoning from the last paragraph of the proof of Statement <ref>. 
 



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