In 1979 Valiant showed that the complexity class VP_e of families with polynomially bounded formula size is contained in the class VP_s of families that have algebraic branching programs (ABPs) of polynomially bounded size. Motivated by the problem of separating these classes we study the topological closure VP_e-bar, i.e. the class of polynomials that can be approximated arbitrarily closely by polynomials in VP_e. We describe VP_e-bar with a strikingly simple complete polynomial (in characteristic different from 2) whose recursive definition is similar to the Fibonacci numbers. Further understanding this polynomial seems to be a promising route to new formula lower bounds. Our methods are rooted in the study of ABPs of small constant width. In 1992 Ben-Or and Cleve showed that formula size is polynomially equivalent to width-3 ABP size. We extend their result (in characteristic different from 2) by showing that approximate formula size is polynomially equivalent to approximate width-2 ABP size. This is surprising because in 2011 Allender and Wang gave explicit polynomials that cannot be computed by width-2 ABPs at all! The details of our construction lead to the aforementioned characterization of VP_e-bar. As a natural continuation of this work we prove that the class VNP can be described as the class of families that admit a hypercube summation of polynomially bounded dimension over a product of polynomially many affine linear forms. This gives the first separations of algebraic complexity classes from their nondeterministic analogs.
@cite_7: 
@cite_15: The determinantal complexity of a polynomial f is defined here as the minimal size of a matrix M with affine entries such that f = det M. This function gives a minoration of the more traditional size of an arithmetical formula. Consider the polynomial “permanent” perm d of a d × d matrix with entries X i,j . A conjecture in complexity theory says that the determinantal complexity (dc) of perm d should not be polynomial in d. In this article we prove that dc(permd)≥d2 2, improving the previously known minoration, 2d. We also begin a systematic study of the function dc, and compute it for the homogeneous polynomials of degree 2.
@cite_21: 
@cite_52: The permanent versus determinant conjecture is a major problem in complexity theory that is equivalent to the separation of the complexity classes VP_ ws and VNP. Mulmuley and Sohoni (SIAM J. Comput., 2001) suggested to study a strengthened version of this conjecture over the complex numbers that amounts to separating the orbit closures of the determinant and padded permanent polynomials. In that paper it was also proposed to separate these orbit closures by exhibiting occurrence obstructions, which are irreducible representations of GL_ n^2 (C), which occur in one coordinate ring of the orbit closure, but not in the other. We prove that this approach is impossible. However, we do not rule out the general approach to the permanent versus determinant problem via multiplicity obstructions as proposed by Mulmuley and Sohoni.
@cite_19: We determine set-theoretic defining equations for the variety of hypersurfaces of degree d in an N-dimensional complex vector space that have dual variety of dimension at most k. We apply these equations to the Mulmuley-Sohoni variety, the GL_ n^2 orbit closure of the determinant, showing it is an irreducible component of the variety of hypersurfaces of degree n in C^ n^2 with dual of dimension at most 2n-2. We establish additional geometric properties of the Mulmuley-Sohoni variety and prove a quadratic lower bound for the determinantal border-complexity of the permanent.
@cite_40: 
@cite_31: Valiant introduced 20 years ago an algebraic complexity theory to study the complexity of polynomial families. The basic computation model used is the arithmetic circuit, which makes these classes very easy to define and open to combinatorial techniques. In this paper we gather known results and new techniques under a unifying theme, namely the restrictions imposed upon the gates of the circuit, building a hierarchy from formulas to circuits. As a consequence we get simpler proofs for known results such as the equality of the classes VNP and VNP"e or the completeness of the Determinant for VQP, and new results such as a characterization of the classes VQP and VP (which we can also apply to the Boolean class LOGCFL) or a full answer to a conjecture in Burgisser's book [Completeness and reduction in algebraic complexity theory, Algorithms and Computation in Mathematics, vol. 7, Springer, Berlin, 2000]. We also show that for circuits of polynomial depth and unbounded size these models all have the same expressive power and can be used to characterize a uniform version of VNP.
@cite_16: We prove that the determinantal complexity of a hypersurface of degree @math d>2 is bounded below by one more than the codimension of the singular locus, provided that this codimension is at least 5. As a result, we obtain that the determinantal complexity of the @math 3×3 permanent is 7. We also prove that for @math n>3, there is no nonsingular hypersurface in @math Pn of degree d that has an expression as a determinant of a @math d×d matrix of linear forms, while on the other hand for @math n≤3, a general determinantal expression is nonsingular. Finally, we answer a question of Ressayre by showing that the determinantal complexity of the unique (singular) cubic surface containing a single line is 5.
@cite_17: Abstract This paper studies geometric properties of the Iterated Matrix Multiplication polynomial and the hypersurface that it defines. We focus on geometric aspects that may be relevant for complexity theory such as the symmetry group of the polynomial, the dual variety and the Jacobian loci of the hypersurface, that are computed with the aid of representation theory of quivers.
Toda @cite_7 proved that several definitions for the class @math are equivalent, see also @cite_31 . In particular @math is the class of polynomials that can be written as determinants of matrices of polynomially bounded size whose entries are affine linear forms. Due to its pure mathematical formulation, lower bounds for this attracted the attention of geometers @cite_15 @cite_19 @cite_16 . Moreover, Mulmuley and Sohoni's geometric complexity approach @cite_21 @cite_40 is also mainly focused on lower bounds for the determinantal complexity and the symmetries of the determinant polynomial play a key role in their work. Recently @cite_52 showed that it is not possible to prove superpolynomial lower bounds on the determinantal complexity using only information about the occurrences non-occurrences of irreducible representations in the coordinate rings of the orbit closures of the determinant and the (padded) permanent. This disproves a major conjecture in geometric complexity theory. The proof in @cite_52 is fairly general and also holds for lower bounds on the formula size. Only very recently the formula size analog to determinantal complexity, the was studied from a geometric perspective @cite_17 .