Points on nodal lines with given direction
Zeev Rudnick Igor Wigman








School of Mathematical Sciences, Tel Aviv University, Tel Aviv,
Israel rudnick@post.tau.ac.il

Department of Mathematics, King's College London, UK
igor.wigman@kcl.ac.uk





We study of the directional distribution function of nodal lines for eigenfunctions of the Laplacian on a planar domain. This quantity counts the number of points  where the normal to the nodal line points in a given direction. We give upper bounds for the flat torus, and compute the expected number for arithmetic random waves.


Points on nodal lines with given direction
    Zeév Rudnick and Igor Wigman
    December 30, 2023
==========================================





§ INTRODUCTION




 §.§ Nodal directions


One of the more intriguing characteristics of a Laplace eigenfunction on a planar domain

is its nodal set. Much progress has been achieved in understanding its length, notably
the work of Donnelly and Fefferman <cit.>, and the recent breakthrough by Logunov and Mallinikova <cit.>, and several researchers have tried to understand the number of nodal domains  (the connected components of the complement of the nodal set),

starting with Courant's upper bound on that number, see <cit.> for the latest result.  
In this note, we propose to study a different  quantity, the directional distribution, measuring an aspect of the curvature of nodal lines.


Let Ω be a planar domain, with piecewise smooth boundary, and let f be an eigenfunction of the Dirichlet Laplacian, with eigenvalue E: -Δ f=Ef.
Given a direction ζ∈ S^1, let N_ζ(f) be the number of points x on the nodal line {x∈Ω: f(x)=0} with normal pointing in the direction ±ζ:

    N_ζ(f)= #{x∈Ω:  f(x)=0, ∇ f(x)/∇ f(x) = ±ζ}.

In particular (<ref>) requires that ∇ f(x)≠ 0, i.e. x is a non-singular point of the nodal line.

In a few separable cases, such as an irrational rectangle, or the disk, one can explicitly compute N_ζ(f): For the irrational rectangle, the nodal line is a grid and N_ζ(f)=0,∞, while for the disk the nodal line is a union of diameters and circles, and we find N_ζ(f)≪√(E) except for O(√(E)) choices of ζ, when N_ζ(f)=∞, see Appendix <ref>. However, in most cases one cannot explicitly  compute N_ζ(f).
The following heuristic suggests that generically the order of magnitude of N_ζ(f) is about E: We expect a “typical" eigenfunction to have an order of magnitude of
E nodal domains <cit.>, and looking at several plots of nodal portraits such as Figure <ref> would lead us to believe that many of the nodal domains are ovals, or at least have a controlled geometry, with O(1) points per nodal domain  with normal parallel to any given direction.
Therefore we are led to expect that the total number of points on the nodal line with normal parallel to ±ζ should be about
E (if it is finite).








To try and validate this heuristic, we study N_ζ(f)  on the standard flat torus =^2/^2 (equivalently taking Ω to be the square, and imposing periodic, rather than Dirichlet, boundary conditions), for both random and deterministic eigenfunctions. We prove deterministic upper bounds, and compute the expected value of N_ζ for “arithmetic random waves" described below.





 §.§ A deterministic upper bound



We want to establish individual upper bounds on N_ζ(f).
Strictly speaking, this is not possible, since there are cases where N_ζ(f)= ∞.


































For instance, the nodal set of the eigenfunctions f(x,y)=sin(2π mx)sin(2π ny) (m,n≥ 1) is a union of straight lines with N_ζ(f)=0 unless ζ=±(1,0),±(0,1) in which case N_ζ(f)=∞.
More generally, one can construct toral eigenfunctions f so that their nodal lines contain a closed geodesic, but also curved components,

see Figure <ref> where we display the eigenfunction

    f(x,y)    =    2(
    sin 8x sin y + sin 7x sin 4y +sin x sin 8y +sin 4xsin 7y )
    
       = 4sin(x)sin(y)(cos x+cos y) h(x,y)

where
 
    h(x,y)=   2 cos (3 x-5 y)-2 cos (2 x-4 y)-2 cos (4 x-4   y)+4 cos (x-3 y)
        +4 cos (3 x-3 y)     
      +2 cos (5 x-3 y)    -4 cos (2 x-2 y)-2 cos (4 x-2 y)
            +6   cos (x-y)+4 cos (3 x-y)+6 cos (x+y)
       +4 cos (3 x+y)
       
    
       -4 cos (2 x+2 y)   -2 cos (4 x+2 y)
      +4   cos (x+3 y)+4 cos (3 x+3 y)
       +2 cos (5 x+3 y)
      -2 cos (2 x+4 y)
      
      -2 cos (4 x+4 y)+2 cos (3 x+5 y)
      -4 cos (2 x)
         +2 cos (6 x)-4 cos (2 y)+2 cos (6 y)-2 .

Theorem <ref> below asserts an upper bound for N_ζ(f) with the only exceptions

being when the nodal line contains a closed geodesic. It will follow as a particular case of a structure result on the set


    _ζ(f) = { x∈Ω:  f(x)=0,  ⟨∇ f(x), ζ^⊥⟩ = 0 }

of “nodal directional points", i.e. the set of nodal points where ∇ f is orthogonal to ζ^⊥ (thus co-linear to ζ).
Note that, by the definition, in addition to the set on the r.h.s. of (<ref>), _ζ(f) contains all
the singular nodal points of f^-1(0), and could also contain certain closed geodesics in direction orthogonal to ζ, as we shall see below.
To state Theorem <ref> we introduce the (standard) notion of “height" for a rational vector.

 


[Height of a rational vector]




  * A rational direction ζ∈^1 is one which is a multiple of an integer vector. Note that ζ is rational if and only if the orthogonal direction ζ^⊥ is rational.


  * For a rational vector ζ∈^1 we denote its height by h(ζ) = max(|k_1|,|k_2|) where (k_1,k_2) is a primitive integer vector (unique up to sign) in the direction of ζ:
  
    ζ = ±(k_1,k_2)/√(k_1^2+k_2^2)  .

  Note that h(ζ)=h(ζ^⊥).







Let ζ∈^1 be a direction, and f be a toral eigenfunction: -Δ f=Ef for some E>0.




  * If ζ is rational, then the set _ζ(f) consists of at
most 
    √(E)/π h(ζ)

closed geodesics orthogonal to ζ,
at most


2/π^2· E nonsingular points not lying on the geodesics, and possibly, singular points of the nodal set.


  * If ζ is not rational, then the set _ζ(f) consists of at most 2/π^2· E nonsingular points,
and possibly, singular points of the nodal set.


  * In particular, if _ζ(f) does not contain a closed geodesic, then

    N_ζ(f) ≤2/π^2· E.






The proof of Theorem <ref>, given in section <ref> below,
is sufficiently robust to apply verbatim to the more general family of
trigonometric polynomials on ^2 of degree ≤√(E).


We note that 
it is possible to construct Laplace eigenfunctions f of arbitrarily high eigenvalues and ζ∈^1 such that
N_ζ(f)=0 vanishes, so that a general lower bound for N_ζ(f) cannot exist. For example,

    f(x,y)=2 cos(2π· m x) + cos(2π· m y)

has eigenvalue E=4π^2m^2 and satisfies N_ζ(f)=0 for ζ = e^iθ with θ near π/2, see
Figure <ref>.






 §.§ Expected number for arithmetic random waves



A better understanding of several properties of nodal lines is obtained if one studies random eigenfunctions.
In 1962, Swerling <cit.> studied statistical properties of contour lines of a general class of planar Gaussian processes, and gave a non-rigorous computation of the expected value of N_ζ for general contour lines,  using the result to bound the number of closed connected components of contour lines.
 We will compute the expected value of N_ζ for

“arithmetic random waves" <cit.>.
These  are random eigenfunctions on the torus,

    f(x) =f_n(x)= ∑_λ∈_nc_λe(⟨λ, x⟩),

where e(z)=e^2π i z and

    _n={λ=(λ_1,λ_2)∈^2:  λ^2=n}

is the set of all representations of the integer n=λ_1^2+λ_2^2
as a sum of two integer squares, and c_λ are standard Gaussian random variables[After understanding the Gaussian case, one may try non-Gaussian ensembles, see e.g. <cit.>.], identically distributed and independent save for the constraint

    c_-λ = c_λ ,

making f_n real valued eigenfunctions of the Laplacian with eigenvalue

    E=4π^2n

for every choice of the coefficients {c_λ}_λ∈_λ
(i.e. for every sample point).

Equivalently f_n:^2→
is a centred Gaussian random field with covariance

    r(x,y)=r_n(y-x) = 1/N_n∑_λ∈_ne(⟨λ,y-x ⟩).

Since r(x,y) depends only on y-x, the random field f_n is stationary, meaning that
for every translation 
    τ_z:f_n(·)↦ f_n(·+z)
 with z∈^2, the law
of τ_z f_n equals the law of f_n:

    τ_z f_nd= f_n.

This, in turn, is equivalent to the law
of the Gaussian multivariate vector (f_n(x_1),…, f_n(x_k)) being equal to the
law of the vector (f_n(x_1+z),…, f_n(x_k+z)) for every x_1,…, x_k∈^2,
z∈^2.

In <cit.> we studied the statistics of the length of the nodal line of f_n.
Since then, very refined data has been obtained on the nodal structure of such random eigenfunctions
(see e.g. <cit.>).


We will compute the expected value of N_ζ for arithmetic random waves. The answer depends on the distribution of lattice points on the circle of radius √(n).
Let μ_n be the atomic measure on the unit circle given by

    μ_n=1/r_2(n)∑_λ∈_nδ_λ/√(n),

where r_2(n):=#_n, and let

    μ_n(k) 
    =1/r_2(n)∑_λ=(λ_1,λ_2)∈_n(λ_1+iλ_2/√(n))^k∈

be its Fourier coefficients.



For ζ=e^iθ∈ S^1, the expected value of N_ζ(f) for the arithmetic random wave (<ref>) is

    [N_ζ] = 1/√(2)n·(1+μ_n(4)·cos(4θ))^1/2.



The statement (<ref>) of Theorem <ref> is valid even if the r.h.s. of (<ref>) vanishes, i.e.
if 
    μ_n(4)·cos(4θ)=-1:
 either

    μ_n = 1/4(δ_± 1+δ_± i)

(“Cilleruelo measure") and θ = ±π/4,±3π/4, or μ_n is the rotation by π/4 of the latter measure
(“tilted Cilleruelo") and ζ is parallel to one of the axes.
These cases are exceptional in the following sense: It is known  <cit.> that for every probability measure μ on the unit circle ^1 there exists a constant c_NS(μ)≥ 0 (the “Nazarov-Sodin constant")
such that if the measures μ_n converge weak-* to μ, 
then the expectation of the number (f_n) of nodal domains of f_n is

    [(f_n)] = (c_NS(μ)+o(1))· n.

Moreover, the Nazarov-Sodin constant c_NS(μ)=0 vanishes, if and only if μ is  one of these exceptional measures

<cit.>. In that case it was shown <cit.> that
most of the nodal components are long and mainly parallel to one of the axes (perhaps, after rotation by π/4);
with accordance to the above, our computation (<ref>) implies in particular that c_NS(μ)=0 for
μ (tilted) Cilleruelo measure, i.e. the “if" part of the aforementioned statement from  <cit.>.












One can study an analogous quantity N_ζ(f) for eigenfunctions on the d-dimensional torus ^d = ^d/^d,
d≥ 3 with eigenvalue 4π^2n. We can establish a result analogous to Theorem <ref> in the higher dimensional
case, showing that for d≥ 3,

    [N_ζ] ∼  C_d n^d/2  ,   n→∞,

for some positive constant C_d>0 independent of ζ, assuming that n≠ 0,4,7  8 if d=3, and n≠ 0 8 if d=4.




 §.§ Acknowledgements

We thank Jerry Buckley, Suresh Eswarathasan, Manjunath Krishnapur, Mark Shusterman and Mikhail Sodin for their comments.
The work was supported  by the European Research Council under the European Union's Seventh
Framework Programme (FP7/2007-2013)/ERC grant agreement n^o 320755 (Z.R.) and  n^o 335141 (I.W.).






§ DETERMINISTIC UPPER BOUND: PROOF OF THEOREM <REF>




Before giving a proof for Theorem <ref> we will need some preparatory results, all related to
the identification of the trigonometric polynomials on ^2 with Laurent polynomials in [z_1,z_2],
via the natural embedding ^2=^1×^1↪^2 (see (<ref>) below).



 §.§ From trigonometric polynomials to (Laurent) polynomials







  * Let  be the space of all complex valued trigonometric polynomials on ^2.
We define an operator Φ:→[z_1,z_2,z_1^-1,z_2^-1] between  and the complex Laurent polynomials
in the following way. For g:^2→ a trigonometric polynomial

    g(x)=∑_λ∈^2
    finite sumc_λe^2π i ⟨λ, x ⟩,

we associate the Laurent polynomial
G=Φ(g)∈[z_1,z_1^-1,z_2,z_2^-1] via the embedding ^2=^1×^1↪^2

    (x_1,x_2)↦ (z_1,z_2)=(e^2π i x_1, e^2π i x_2),

or, explicitly,

    G(z) = g(x) = ∑_λ∈^2c_λz^λ,

where for z=(z_1,z_2)∈^2 and λ∈_n we denote z^λ:=z_1^λ_1· z_2^λ_2.


  * For k=1,2 let D_k:[z_1,z_2] →[z_1,z_2] be the operator

    D_k:p(z)↦ z_k∂ p(z)/∂ z_k .



  * For ξ∈^1 denote the operator

    D_ξ = ⟨ (D_1,D_2), ξ⟩ = ξ_1D_1+ξ_2D_2.










The following properties are immediate from the definitions:








  * 

For every ξ∈^1 the operator D_ξ (in particular, D_1 and D_2) is a derivation, i.e. it is a linear operator
satisfying the Leibnitz law

    D_ξ(p(z)q(z)) = D_ξp(z)· q(z)+p(z)· D_ξq(x).



  * For every g, a trigonometric polynomial as in (<ref>), and x=(x_1,x_2)∈^2, we have

    g(x) = (Φ g)(z) = G(z),

where z=z(x) is given by (<ref>) and G=Φ g.


  * For ξ∈^1, if G=Φ g, then

    1/2π iΦ(∂_ξg) = D_ξG,

i.e. if under Φ, g maps to g↦G, then its (normalised) directional derivative
1/2π i∂_ξg maps to D_ξG.









 §.§ Auxiliary lemmas




Let g:^2→ be

a trigonometric polynomial
(<ref>), x_0∈ g^-1(0) a nonsingular zero,
G=Φ(g)∈[z_1,z_1^-1,z_2,z_2^-1],
and G(z)=z^δG(z) ∈[z_1,z_2], so that

    G(z_0) = g(x_0) = 0,
 where z_0=z(x_0)∈^2 is the point corresponding to x_0
via (<ref>). Suppose also that P| G is an irreducible factor of G such that P(z_0) = 0.
Then P^2∤ G.





Assume by contradiction that, under the assumptions of Lemma <ref>, we have that

    P^2| G

we then claim that
in this case necessarily ∇ g(x_0) = 0, contradicting the non-singularity of x_0 as a zero
of g. We show that ∂ g/∂ x_k(x_0) = 0,  k=1,2.

Since D_k is a derivation in [z_1,z_2], and by (<ref>) we have that

    Φ(1/2π i∂ g/∂_x_k) = D_kG = D_k(z^-δG) = z^-δ· D_kG +
    G· D_kz^-δ.

Since both G and D_kG are divisible by P by our assumption (<ref>), we have
G(z_0)=D_kG(z_0) = 0. Substituting this into (<ref>), and bearing in mind (<ref>), this yields that
∂ g/∂ x_k(x_0) = 0.
Thus x_0 is a singular zero of g, contradicting our assumption.




Let G∈[z_1,z_1^-1,z_2,z_2^-1] be a Laurent polynomial,
δ∈^2_≥ 0 so that

    G(z)=z^δG(z) ∈[z_1,z_2]

is a polynomial, with δ minimal in  the sense that z_j∤ G. Let
Q_ξ(z) = D_ξ(G)(z)
and

    Q_ξ(z):=z^δ·Q_ξ(z)∈[z_1,z_2].

Suppose that

    P|(G,Q_ξ)

is an irreducible polynomial, such that P^2∤ G. Then necessarily
D_ξP is a scalar multiple of P, i.e. there exists t∈ so that

    D_ξP = t· P.





First, since by Lemma <ref>, D_ξ is a derivation, we have that

    D_ξG    = D_ξ(z^δ· z^-δG) = D_ξ(x^δ·G) =
    D_ξ(x^δ)·G + x^δ· D_ξ(G) 
       = ⟨δ,ξ⟩ x^δ·G
    +x^δQ_ξ = ⟨δ,ξ⟩ G + Q_ξ,

by (<ref>) and (<ref>).
Hence, since, by assumption (<ref>), both summands on the r.h.s. of (<ref>) are divisible by P, so is D_ξG, i.e.

    P| D_ξG.


Now let us write

    G=P· A

for some A∈[z_1,z_2]; since by assumption P is irreducible, and P^2∤ G by
Lemma <ref>, this necessarily implies

    (P,A)=1.

Applying the derivation D_ξ on (<ref>)
we obtain:

    D_ξG = D_ξ(P)· A+P· D_ξA,

which, together with (<ref>) yields that

    P| D_ξ(P)· A,

which, by (<ref>), forces

    P| D_ξ(P).

Note that if

    P(z)=∑_α∈_≥ 0^2p_αz^α

is a finite sum, then

    D_ξ(P)(z)=∑_α∈_≥ 0^2⟨ξ,α⟩ p_αz^α

is of degree at most  the degree of P. Hence (<ref>) implies that
D_ξP is a scalar multiple of P.





Let ξ∈^1, t∈, and P∈[z_1,z_2] nonconstant irreducible polynomial such that
z_1,z_2∤ P, and

    D_ξP = t· P.

Then the following hold:



  * The direction ξ is rational (i.e. the vector ξ is a multiple of a rational vector).


  * The polynomial P is necessarily of the form

    P(z) = p_1z_1^k_1+p_2z_2^k_2

for some p_1,p_2∈\{0}, and (k_2,k_1)∈_≥ 0^2 is a primitive vector (unique up to sign) satisfying

    (k_2,k_1)/(k_2,k_1) = ±ξ.








Writing P as a finite sum

    P(z) = ∑_α⟨ξ,α⟩ p_αz^α,

(the finite sum over α∈_≥ 0^2), the equality (<ref>) is equivalent to

    ⟨ξ,α⟩· p_α=t· p_α

for every α∈_≥ 0^2, i.e.

    ⟨ξ,α⟩ = t

for every α∈_≥ 0^2 with p_α 0. Note that P is not a monomial
(as otherwise P would be divisible by either z_1 or z_2), hence (<ref>)
is valid for at least two distinct α. Therefore, for these α, one has

    ⟨ξ,α-α'⟩=0,

which forces ξ to be rational, i.e. yields the first statement of Lemma <ref>.




Now assume that the rational vector ξ=u/u is a multiple of a primitive integer vector u∈^2. We may
then rewrite (<ref>) as

    ⟨ u,α⟩ = s,

with s=u· t, uniquely determined by ξ and t, and to have any solution to (<ref>), necessarily s∈.
The integer solutions to (<ref>), considered as an equation in α, are

    α=α^0+k· v,

where α^0 is a particular solution to (<ref>),
and v∈^2 is the primitive integer vector orthogonal to u, unique up to sign, some of whose coordinates might be negative.

Note that 
    ζ =  v/v

is a unit vector orthogonal to ξ.

Since the collection 
    {α∈^2:  p_α 0}
 is finite (corresponding to a finite collection of k in
(<ref>)), we can choose α^0 a particular solution of
(<ref>) so that

    p_α^0 0,

and the numbers k in (<ref>) satisfy 0≤ k≤ K for some
K>0; by (<ref>) we necessarily have α_0∈_≥ 0 ^2. We may then write:

    P(z) = ∑_k=0^Kp_kz^α^0+k· v = z^α^0·∑_k=0^K(z^v)^k =
    z^α^0· Q(z^v),

where Q(w)∈[w] is a (one variable) complex polynomial, which, by above, is not a monomial.

We claim that the irreducibility of P implies the irreducibility of Q, which, in turn, implies that Q is linear.
For if Q were reducible, we could write

    Q(w)=A(w)· B(w)

for some nonconstant polynomials A,B∈[w]. Substituting (<ref>) into (<ref>), we obtain

    P(z) = z^α^0· A(z^v)B(z^v).

As one or both components of v might be negative, (<ref>)
does not immediately imply that P is reducible. Write

    A(z^v) = z^-α^1A(z),


    B(z^v) = z^-α^2B(z),

where α^1,α^2∈_≥ 0^2 are minimal so that A(z), B(z)∈[z] are polynomial,
so that A,B are not divisible by z_1,z_2.
We then have

    P(z) = z^α^0-α^1-α^2·A(z)B(z).

Since P is not divisible by z_1,z_2 and neither are A and B, the equality (<ref>) implies that
α^0-α^1-α^2 = 0, so that

    P(z)=A(z)·B(z)

is a factorization of P into nonconstant polynomials, contradicting the assumption that P is irreducible, and hence
Q as in (<ref>) is itself irreducible in [w], so

    Q(w)=q_0+q_1w

with q_0,q_1∈^*, is linear.

Substituting  (<ref>) into (<ref>) gives

    P(z) = q_0z^α_0 + q_1z^α_0+v,

and α_0,α_0+v ∈_≥ 0 ^2. Since α_0α_0+v and z_1,z_2∤ P,
the form (<ref>) of P reduces to (<ref>), and it also forces

    v = (-k_1,k_2),

hence (k_2,k_1) is a primitive lattice point of ^2, co-linear with ξ.





 §.§ Proof of Theorem <ref>




Let f=f_n be 
a

toral eigenfunction (<ref>) (it is a monochromatic trigonometric polynomial whose frequency set
_n is given by (<ref>)), and

    G=Φ(f)∈[z_1,z_1^-1,z_2,z_2^-1]
 be the Laurent polynomial associated to f as in
Lemma <ref>, so that

    G(z) = f(x) = ∑_λ∈_nc_λz^λ .

Note that for λ∈_n, we have |λ_1|+λ_2|≤√(2n).
To make G into a polynomial in [z_1,z_2] we multiply G by a monomial z^δ with δ∈_≥ 0^2 satisfying

    δ_1+δ_2≤√(2n),

to write

    G(z)=z^δG(z),

with δ minimal, so that, in particular, G(z) is not divisible by z_1  or z_2.
By (<ref>) and (<ref>),
we have


    (G) ≤ 2√(2)·√(n).

Now let Q_ξ=1/2π iΦ(∂_ξf) be the Laurent polynomial corresponding to the directional
derivative ∂_ξf(x) of f where ξ=ζ^⊥ is orthogonal to ζ. By Lemma <ref> we have

    Q_ξ(z) = D_ξ(G(z)) = 1/2π iΦ(∂_ξf)(z)=
    ∑_λ∈_n⟨λ,ξ⟩ c_λz^λ,

and

    Q_ξ(z):=z^δ·Q_ξ(z)∈[z_1,z_2]

with δ same as in (<ref>), is a polynomial of degree

    (Q_ξ)≤ 2√(2)·√(n),

though might be divisible by z_1 or z_2.
By (<ref>), (<ref>), (<ref>),
and (<ref>), for some x_0∈^2 we have

    f(x_0)=∂_ξf(x_0)=0,
 (without imposing ∇ f(x_0) 0), if and only if z_0=z(x_0) is a joint zero of both G and Q_ξ, i.e. 
    G(z_0)=Q_ξ(z_0) = 0.


Now let

    D= (G,Q_ξ)

be the greatest common divisor of G and Q_ξ

    G(z) = A(z)· D(z)

and

    Q_ξ(z) = B(z)· D(z),

where

    (A,B) = 1

and

    (D), (A)≤(G)≤ 2√(2)·√(n),    (B)≤(Q_ξ) ≤ 2√(2)·√(n)

by (<ref>) and (<ref>), and,
by the above, we are interested in z=(z_1,z_2)∈^2, so that |z_1|=|z_2|=1
and G(z)=Q_ξ(z) = 0.



Given z_0∈^2 we have that G(z_0)= Q_ξ(z_0) = 0,
if and only if either

    A(z_0)=B(z_0) = 0,

or D(z_0)=0 (both cannot occur simultaneously).
Denote

    ^1(G,Q_ξ) := { z∈^2:  A(z) = B(z) = 0}

and

    ^2(G,Q_ξ) := { z∈^2:  D(z) = 0},

the nodal directional points of the first and second type respectively.
The meaning of the above is that, under the embedding (<ref>)
of ^1×^1⊆^2,

    {x∈^2; f(x)= ⟨∇ f, ξ⟩ = 0}↦ (^1(G,Q_ξ) ∪^2(G,Q_ξ))∩^1×^1.

Hence understanding of 
    ^1(G,Q_ξ) ∪^2(G,Q_ξ)
 will also allow for bounding the size of
the l.h.s. of (<ref>); note that, unlike the definition (<ref>)
of N_ζ, the l.h.s. of (<ref>)
includes singular points of f^-1(0), having no bearing on giving an upper bound for N_ζ
via one for the r.h.s. of (<ref>).
Since A and B are co-prime by (<ref>), and bearing in mind (<ref>) and the definition
(<ref>), it follows that ^1(G,Q_ξ)
consists of finitely many isolated points, and its cardinality is bounded, by Bézout's Theorem[which states that if A,B∈[z_1,z_2] are co-prime polynomials, then the number of common zeros of A and B is bounded by A· B.]

    |^1(G,Q_ξ)| ≤ (A)·(B) ≤ 8n = 2E/π^2,

on using (<ref>) and (<ref>).

Now we turn to understanding ^2(G,Q_ξ) as in (<ref>). Let P|D be an irreducible divisor of D=(G,Q_ξ),
and let x_0∈_ζ(f)∈^2 be a nonsingular nodal directional point so that P(z_0) = 0, where z_0=z(x_0),
the map in (<ref>). Then, thanks to Lemma <ref>, P^2∤ G, so that we may apply Lemma
<ref> to deduce that

    D_ξP = t· P,

for some scalar t∈. By invoking Lemma <ref>, the equality (<ref>)
in turn implies that ξ is a rational direction, and

    P(z) = p_1z_1^k_1+p_2z_2^k_2,

where the primitive vector

    (k_2,k_1)∈^2_≥ 0 is co-linear to ξ = ζ^⊥,   i.e.  orthogonal  to ζ.


Thus

    D = (∏_i=1^K P_j(z))· E(z),

where for every j=1,2,… K the polynomial P_j is of the form

    P_j(z) = p_1;jz_1^k_1+p_2;jz_2^k_2,

for some p_1;j,p_2;j∈, and
E(z) is the product of irreducible factors P| D of D so that P^2| D (corresponding to the singular points x_0∈^2),
and those irreducible P| D that don't vanish on ^1×^1⊆^2. It then follows that

    K ≤(D)/max(k_1,k_2)≤ 2√(n)/h(ξ)=2√(n)/h(ζ)

by (<ref>).

Now using (<ref>) on
(<ref>), (<ref>),
we have that, under the embedding (<ref>), the zeros of D(z) correspond to the zeros of

    d(x):=D(z(x)) = (∏_j=1^K(p_1;je^2π i k_1 x_1 +p_2;je^2π i k_2x_2)) ·Ẽ(x),

where Ẽ(x):^2→ is the trigonometric polynomial corresponding to E(z)=Ẽ(x), that only has singular zeros.
Let

    h_j(x):= p_1;je^2π i x_1k_1 +p_2;je^2π i x_2k_2

be a   factor of (<ref>); by construction (<ref>)
we know a priori that the zero locus of h_j on ^2 is non-empty. In this case, necessarily
|p_1;j|=|p_2;j|, and upon writing 
    -p_2;j/p_1;j =: e^2π i φ
 for φ∈ [0,1), the zero locus
of h_j is given by

    h_j^-1(0)    = {(x_1,x_2)∈^2:   e(x_1k_1-x_2k_2) = -p_2;j/p_1;j}
       =
    {(x_1,x_2)∈^2:   x_1k_1-x_2k_2 = φ1},

hence is a closed geodesic in ^2 (it has a single connected component, since, by assumption,
(k_1,k_2)=1), orthogonal to (k_1,-k_2), of length √(k_1^2+k_2^2), and, recalling (<ref>),
the geodesic h_j^-1(0) is orthogonal to ζ.
In summary, under the embedding (<ref>), the nonsingular points on f^-1(0) corresponding to
the set ^2(G,Q_ξ)∩ (^1×^1) consist of 
    ≤ 2√(n)/h(ξ)=√(E)/h(ζ)

closed geodesics orthogonal to ζ, concluding the statement of Theorem <ref>.






§ EXPECTED NODAL DIRECTION NUMBER FOR ARITHMETIC RANDOM WAVES: PROOF OF THEOREM <REF>



In this section, we compute the expected value of N_ζ for arithmetic random waves. The formal computation is along the lines of Swerling's paper <cit.>, but his argument relied on several assumptions, some implicit, on the nature of the relevant Gaussian field, which are difficult to isolate and check separately. Thus we carry out the computation ab initio.




 §.§ Proof of Theorem <ref>






Let ξ = ζ^⊥ be the orthogonal vector to ζ, and define,

    N_ζ(f) = #{x∈^2:  f(x)=⟨∇ f(x),ξ⟩ = 0  }

to be the size of the set _ζ(f) in (<ref>), finite or infinite.
Equivalently,

    N_ζ(f) = N_ζ(f) + #(f),

where

    (f) = { x∈^2:  f(x)=0, ∇ f(x) = 0 }

is the set of singular nodal points of f.

Since by Bulinskaya's Lemma  <cit.>, the singular set (f)
is empty almost surely (that the statement of Bulinskaya's Lemma is valid in our concrete case
was established in  <cit.>), we have that

    N_ζ(f) = N_ζ(f) = #{x∈^2:  f(x)=⟨∇ f(x),ξ⟩ = 0  }.

That is, upon defining the Gaussian random field G:^2→^2

    G(x)=G_ξ(x) = (f(x),⟨∇ f(x),ξ⟩),

then N_ζ equals almost surely the number of zeros of G. Let J_G(x) be the Jacobian of G given by

    J_G(x)    = (
    f_1   f_2
     f_11ξ_1+f_12ξ_2    f_12ξ_1+f_22ξ_2) = f_1(f_12ξ_1+f_22ξ_2) - f_2(f_11ξ_1+f_12ξ_2),

where we denote f_i = ∂ f/∂ x_i, f_ij = ∂^2 f/∂ x_i∂ x_j,
and all the derivatives of f are evaluated at x.

The zero density function is

    K_1(x)=K_1;ξ(x) = ϕ_G(x)(0,0)·[|J_G(x)| | G(x)=0],

where ϕ_G(x) is the probability density function of the random vector G(x)∈^2;
by the aforementioned stationarity (<ref>) of f_n, we have

    K_1(x)≡ K_1(0).

By Kac-Rice  <cit.> and (<ref>), we have that

    [N_ζ] = ∫_^2 K_1(x)dx,

provided that the distribution of G(x) is non-degenerate for every x∈^2. By stationarity, it is  sufficient to check non-degeneracy of  G(0), which is valid since (f(0),∇ f(0)) ∈^3 is non-degenerate by the computation below.
The statement of Theorem <ref> follows upon substituting the statements of Lemma <ref> and Proposition
<ref> below into (<ref>) so that

    K_1(x) = 1/√(2)n·(1+μ_n(4)·cos(4θ))^1/2,

and then finally into (<ref>).



In course of the proof of Theorem <ref> we used the following results established in
<ref> below:



Let G:^2→^2 be the Gaussian field defined by (<ref>),
and ϕ_G(x) the probability density function of G(x). Then for every x∈^2 we have

    ϕ_G(x)(0,0) = 1/2π√( C_G(x)) = 1/2^3/2π^2√(n).






Let G:^2→^2 be the Gaussian field defined by (<ref>), and J_G(x) its Jacobian. Then
 the conditional expectation of |J_G(x)| conditioned on G=0 is

    [|J_G(x)||G(x)=0] = 2 π^2(1+μ_n(4)·cos(4θ))^1/2· n^3/2






 §.§ Proofs of Lemma <ref> and Proposition <ref>: evaluating the zero density






  The covariance matrix of (f(x),∇ f(x)) was computed in <cit.> to be

    C_(f,∇ f) = (
    1 
       2π^2nI_2),

in particular f(x) is independent of ∇ f(x);
hence the covariance matrix of G is

    C_G(x) = (
    1 
       2π^2n
    ),

where we used

    (⟨∇ f(x),ξ⟩) = ξ_1^2(f_1)+ξ_2^2(f_2) =
    2π^2n,
 since 
    ξ_1^1+ξ_2^2=1.
 Thus

    ϕ_G(x)(0,0) = 1/2π√( C_G(x)) = 1/2^3/2π^2√(n).






We are going to work under the assumption ξ_2 0; one can easily see that the same result
holds for ξ_2 = 0 true, e.g. by switching between ξ_1 and ξ_2; by stationarity we may assume x=0.
Since f is a Laplace eigenfunction of eigenvalue 4π^2n, we have that[This fact helps in simplifying the computation of the first intensity by allowing us to reduce the size of the covariance matrix, as seen in  another few steps.]

    f(x) = - 1/4π^2n(f_11+f_22),

and therefore

    [|J_G(x)|| G(x)=0] = [|J_G(x)| | f_11+f_22=0, f_1ξ_1+f_2ξ_2=0].

Then (recall that we assumed ξ_2 0)

    [|J_G(x)||G(x)=0] 
       = [|J_G(x)| | f_11+f_22=0, f_1ξ_1+f_2ξ_2=0]
    
       =
    [| f_1(f_12ξ_1+f_22ξ_2) - f_2(f_11ξ_1+f_12ξ_2)|  | f_11+f_22=0,  f_1ξ_1+f_2ξ_2=0]
    
       = [| f_1(f_12ξ_1-f_11ξ_2) - f_2(f_11ξ_1+f_12ξ_2)|  | f_11+f_22=0,  f_1ξ_1+f_2ξ_2=0]
    
       =[| f_1·( f_12ξ_1-f_11ξ_2 + f_11ξ_1^2/ξ_2+f_12ξ_1)| | f_11+f_22=0,  f_1ξ_1+f_2ξ_2=0 ]
    
       =
    [| f_1·( 2f_12ξ_1-f_11ξ_1^2-ξ_2^2/ξ_2)| | f_11+f_22=0,  f_1ξ_1+f_2ξ_2=0 ].

Hence we are interested in the distribution of (f_1,f_11,f_12)(x)
conditioned on

    f_11+f_22=0,  f_1ξ_1+f_2ξ_2=0.

The covariance matrix of (f_1,f_2,f_11,f_12,f_22) is given by Lemma <ref>,
we will then compute the covariance matrix
of (f_1,f_11,f_12,f_1ξ_1+f_2ξ_2,f_11+f_22), and then condition on
the last two variables.
To avoid carrying on the constants we transform the variables

    ((X_1,X_2),(X_3,X_4,X_5)) = (1/(2π^2)^1/2√(n)(f_1,f_2), 1/(2π^4)^1/2n(f_11,f_12,f_22))

with covariance matrix

    C_X_1,…, X_5 =
    ( 
    I_2× 2   
     0    A_3× 3),

with

    A = ( 
    3+μ_n(4)    0    1- μ_n(4)   
    
    0    1- μ_n(4)    0 
    
    1- μ_n(4)    0    3+μ_n(4),
    )

and we are to compute

    [|J_G(x)| | G(x)=0] =
    
    2π^3n^3/2·   E[| X_1·( 2X_4ξ_1-X_3ξ_1^2-ξ_2^2/ξ_2)| | X_3+X_5=0,  X_1ξ_1+X_2ξ_2=0 ].


Next we compute the covariance matrix of (X_1,X_3,X_4,X_3+X_5,X_1ξ_1+X_2ξ_2)
to be

    C_(X_1,X_3,X_4,X_3+X_5,X_1ξ_1+X_2ξ_2) =
    (
    B_3× 3   D_3× 2
     D^t_2× 3   E_2× 2),

where B=C_X_1,X_3,X_4 is the covariance matrix of (X_1,X_3,X_4),
E=C_X_3+X_5,X_1ξ_1+X_2ξ_2 is the covariance matrix of 
    (X_3+X_5,X_1ξ_1+X_2ξ_2),

and 
    D=[(X_1,X_3,X_4)^t(X_3+X_5,X_1ξ_1+X_2ξ_2)].

From the above it follows directly that

    B = (
    1       
       3+μ_n(4) 
          1- μ_n(4)
    ),


    D = (
    0    ξ_1
     4     0 
     0    0
    ),


    E=( 
    8     0 
     0    1
    ).


Let Y=(Y_1,Y_2,Y_3) be the vector (X_1,X_3,X_4) conditioned on

    X_3+X_5=X_1ξ_1+X_2ξ_2=0,
 so that under the new notation (<ref>) is

    [|J_G(x)| | G(x)=0] 
       = 2π^3n^3/2[| Y_1·( 2Y_3ξ_1-Y_2ξ_1^2-ξ_2^2/ξ_2)|].

The covariance matrix of Y is

    C_Y=B-DE^-1D^t   = (
    1-ξ_1^2
       1+ μ_n(4) 
          1- μ_n(4)
    ) 
       =  (ξ_2^2
       1+ μ_n(4) 
          1- μ_n(4)
    ),

where for the above we computed

    DE^-1D^t   = (
    0    ξ_1
     4     0 
     0    0
    ) ·( 1/8
        1
    ) ·( 
    0    4    0 
    ξ_1    0    0 )
    
       = ( 
    0    ξ_1
    1/2    0 
     0    0
    ) ·( 
    0    4    0 
    ξ_1    0    0) = ( ξ_1^2
       2 
           0
    )


We may simplify the expression (<ref>) using the fact that the Y_j are independent:

    [|J_G(x)| | G(x)=0]    = 2π^3n^3/2[|Y_1|]
    ·[|2Y_3ξ_1-Y_2ξ_1^2-ξ_2^2/ξ_2|]
    
       = 2^1/2π^5/2n^3/2·[| ( 2Y_3ξ_1ξ_2-Y_2(ξ_1^2-ξ_2^2))|]
    
       = 2^1/2π^5/2n^3/2·[ |Z_1sin(2θ)+Z_2cos(2θ)   | ],

where (Z_1=Y_3,Z_2=-Y_2) is a centered Gaussian with covariance

    (
    1-μ_n(4) 
       1+ μ_n(4)
    )  ,
 also valid for ξ_2=0
(θ is the direction of ζ, or of ξ by the sign invariance of the distribution of Z_1, Z_2).

The random variable 
    A:=Z_1sin(2θ)+Z_2cos(2θ)
 is centered Gaussian, whose variance is

    (A) = (1- μ_n(4))sin(2θ)^2 + (1+ μ_n(4))cos(2θ)^2 = 1+μ_n(4)cos(4θ),

and (<ref>) is

    [|J_G(x)| | G(x)=0]    = 2^1/2π^5/2n^3/2·[|A|]
        = 2^1/2π^5/2n^3/2·√(2/π)√((A))
       = 2 π^2n^3/2· (1+μ_n(4)cos(4θ))^1/2,

which is the statement of Proposition <ref>.





 §.§ Auxiliary lemmas




We have

    1/N∑_λ∈_nλ_1^4 = n^2(3/8+1/8μ_n(4)),

and

    1/N∑_λ∈_nλ_1^2λ_2^2 =
    n^2/8(1- μ_n(4)  ) .






Let f=f_n be the arithmetic random waves (the random field (<ref>) where c_λ are
assumed to be i.i.d. standard Gaussian
save to (<ref>)), and
X=(f_1,f_2,f_11,f_12,f_22) vector of various derivatives evaluated at x=0. Then
X is centered multivariate Gaussian with covariance matrix

    C_f_1,f_2,f_11,f_12,f_22
       =
    ( 
    2π^2n    0    0    0    0 
    
    0     2π^2n    0    0    0 
    
    0    0    2π^4n^2(3+μ_n(4))    0    2π^4 n^2(1- μ_n(4)  ) 
    
    0    0    0    2π^4n^2(1- μ_n(4))    0 
    
    0    0    2π^4 n^2(1- μ_n(4))    0    2π^4n^2(3+μ_n(4))
    ).






Recall that the covariance function of f_n is given  by (<ref>).
We have, using the symmetries,

    [f_1(x)^2]=-r_11(0)=[f_2(x)^2] = 2π^2n


    [f_1(x)f_2(x)]=-r_12(0) = 0,


    [f_1(x)f_11(x)] =-r_111(0)= 0,


    [f_1(x)f_12(x)]=-r_112(0)=0,


    [f_11(x)^2]   =[f_22(x)^2] =r_1111(0) = 16π^4/N∑_λ∈_nλ_1^4
       = 16π^4n^2(3/8+1/8μ_n(4)) = 2π^4n^2(3+μ_n(4))

by Lemma <ref>, and

    [f_12(x)^2]   = [f_11(x)f_22(x)] =
    16π^4/N∑_λ∈_nλ_1^2λ_2^2
     = 2π^4 n^2(1- μ_n(4)  ).








§ SEPARABLE DOMAINS


We describe some  cases when the nodal sets, hence N_ζ(f), can be explicitly computed.



 §.§ Irrational rectangles

Take a rectangle with width π/√(α) and height π, with aspect ratio √(α), and assume that α is irrational.
Then the eigenvalues of the Dirichlet Laplacian consist  of the numbers α m^2+n^2 with integers m,n≥ 1, and the corresponding eigenfunctions are

    f_m,n(x,y) =sin(√(α)mx)sin(ny)  .

The nodal lines consist of a rectangular grid, and one has N_ζ(f_m,n)=0  or ∞.



 §.§ The unit disk

Let Ω = {|x|≤ 1} be the unit disk, and (r,θ) be polar coordinates.
The eigenfunctions of the Dirichlet Laplacian are

    f_m,k(r,θ) = J_m(j_m,kr)cos(mθ+ϕ)

where J_m(z) is the Bessel function, with zeros {j_m,k:k≥ 1}, and ϕ∈ [0,2π) is arbitrary. The corresponding eigenvalue is

    E=j_m,k^2.

In particular, for m≥ 1 the eigenspaces have dimension two.

We will need McCann's inequality <cit.>

    j_m,k^2 ≥π^2(k-1/4)^2 +m^2.

For m=0 (the radial case), the eigenfunctions are f_0,k(r,θ) =J_0(j_0,kr), 0≤ r≤ 1, and have k-1 interior nodal lines, which are the concentric circles r=j_0,ℓ/j_0,k, ℓ=1,…, k-1. Thus for any direction ζ∈ S^1, we have

    N_ζ(f_0,k) = 2(k-1).


For m≥ 1, the nodal line of the eigenfunction f_m,k is a union of the m diameters cos(mθ+ϕ)=0 and k-1 concentric circles r=j_m,ℓ/j_m,k, ℓ=1,…,k-1 (for k=1 there are only diameters), see Figure <ref>. Thus there are 2m values of ζ where N_ζ(f_m,k)=∞, and for all other directions we have

    N_ζ(f_m,k)=2(k-1).

Using

McCann's inequality (<ref>), and (<ref>), the above yields that for m≥ 0, k≥ 1, except for 2m≤ 2√(E) directions where N_ζ(f_m,k)=∞, we have

    N_ζ(f_m,k) ≤2/π√(E) .




99


AW
J.-M. Azaïs, M. Wschebor, Level sets and extrema of random
processes and fields, John Wiley & Sons Inc., Hoboken, NJ, 2009.




Bourgain Pleijel
J. Bourgain, On Pleijel's nodal domain theorem.
Int. Math. Res. Not. IMRN 2015, no. 6, 1601–1612.





CNNV
M.-C. Chang, H. Nguyen, O. Nguyen and V. Vu.
Random eigenfunctions on flat tori: universality for the number of intersections. 
	arXiv:1707.05255 [math.PR]

Donnelly-Fefferman
H. Donnelly, and C. Fefferman Nodal sets of eigenfunctions on
Riemannian manifolds, Invent. Math. 93 (1988), 161–183.











KW
P. Kurlberg and I. Wigman,
Variation of the Nazarov-Sodin constant, available online <https://arxiv.org/abs/1707.00766>. Announcement
C. R. Math. Acad. Sci. Paris 353 (2015), no. 2, 101–104.

K-K-W 
M. Krishnapur, P. Kurlberg and I. Wigman. Nodal length fluctuations for arithmetic random waves, Ann. Math. (2) 177, no. 2, 699–737 (2013).


Logunov2
A.  Logunov Nodal sets of Laplace eigenfunctions: polynomial upper estimates of the Hausdorff measure.
Annals of Math.,    187 (2018), Issue 1 221–239. 	arXiv:1605.02587 [math.AP]

Logunov1
A.  Logunov Nodal sets of Laplace eigenfunctions: proof of Nadirashvili's conjecture and of the lower bound in Yau's conjecture.
Annals of Math.,    187 (2018), Issue 1 241–262.	arXiv:1605.02589 [math.AP]

LM
A.  Logunov  and E.  Malinnikova
Nodal sets of Laplace eigenfunctions: estimates of the Hausdorff measure in dimension two and three, 	arXiv:1605.02595 [math.AP]

MPRW
D. Marinucci, G. Peccati, M.  Rossi and I. Wigman, Non-universality of nodal length distribution for arithmetic random waves.
Geom. Funct. Anal. 26 (2016), no. 3, 926–960.

McCann
R.C. McCann.
Lower bounds for the zeros of Bessel functions.
Proc. Amer. Math. Soc. 64 (1977), no. 1, 101–103.

PR
G.  Peccati and M.  Rossi.
Quantitative limit theorems for local functionals of arithmetic random waves.
	arXiv:1702.03765 [math.PR]


Sodin-Nazarov
F. Nazarov and M.  Sodin, On the number of nodal domains of random spherical harmonics. Amer. J. Math. 131 (2009), no. 5, 1337–1357.

NS 
Nazarov, F.; Sodin, M.
Asymptotic laws for the spatial distribution and the number of connected components of zero sets of Gaussian random functions.
Zh. Mat. Fiz. Anal. Geom. 12 (2016), no. 3, 205–278.

ORW2007
F. Oravecz, Z. Rudnick and I. Wigman The Leray measure of nodal sets for random eigenfunctions on the torus.
Annales de l'institut Fourier, Volume 58 (2008) no. 1 , p. 299–335





Rozenshein
Y. Rozenshein, The Number of Nodal Components of Arithmetic Random Waves. Int. Math. Res. Not. IMRN 2017, no. 22, 6990–7027.  arXiv:1604.00638 [math.CA]


RW2008
Z. Rudnick and I. Wigman.
On the volume of nodal sets for eigenfunctions of the Laplacian on the torus. Ann. Henri Poincaré 9 (2008), no. 1, 109–130.




Swerling
P. Swerling. Statistical properties of the contours of random surfaces. IRE Trans. IT-8 1962 315–321.






