We present polynomial families complete for the well-studied algebraic complexity classes VF, VBP, VP, and VNP. The polynomial families are based on the homomorphism polynomials studied in the recent works of Durand et al. (2014) and Mahajan et al. (2018). We consider three different variants of graph homomorphisms, namely injective homomorphisms , directed homomorphisms , and injective directed homomorphisms , and obtain polynomial families complete for VF, VBP, VP, and VNP under each one of these. The polynomial families have the following properties: • The polynomial families complete for VF, VBP, and VP are model independent, i.e., they do not use a particular instance of a formula, algebraic branching programs, or circuit for characterising VF, VBP, or VP, respectively. • All the polynomial families are hard under p -projections.

No abstract

Schur Polynomials are families of symmetric polynomials that have been classically studied in Combinatorics and Algebra alike. They play a central role in the study of Symmetric functions, in Representation theory [Sta99], in Schubert calculus [LM10] as well as in Enumerative combinatorics [Gas96, Sta84, Sta99]. In recent years, they have also shown up in various incarnations in Computer Science, e.g, Quantum computation [HRTS00, OW15] and Geometric complexity theory [IP17].However, unlike some other families of symmetric polynomials like the Elementary Symmetric polynomials, the Power Symmetric polynomials and the Complete Homogeneous Symmetric polynomials, the computational complexity of syntactically computing Schur polynomials has not been studied much. In particular, it is not known whether Schur polynomials can be computed efficiently by algebraic formulas. In this work, we address this question, and show that unless every polynomial with a small algebraic branching program (ABP) has a small algebraic formula, there are Schur polynomials that cannot be computed by algebraic formula of polynomial size. In other words, unless the algebraic complexity class VBP is equal to the complexity class VF, there exist Schur polynomials which do not have polynomial size algebraic formulas.As a consequence of our proof, we also show that computing the determinant of certain generalized Vandermonde matrices is essentially as hard as computing the general symbolic determinant. To the best of our knowledge, these are one of the first hardness results of this kind for families of polynomials which are not multilinear. A key ingredient of our proof is the study of composition of well behaved algebraically independent polynomials with a homogeneous polynomial, and might be of independent interest.

No abstract

In this paper we prove the following two results.-We show that for any C ∈ {mVF, mVP, mVNP}, C = C. Here, mVF, mVP, and mVNP are monotone variants of VF, VP, and VNP, respectively. For an algebraic complexity class C, C denotes the closure of C. For mVBP a similar result was shown in [4]. Here we extend their result by adapting their proof. -We define polynomial families {P(k)n} n≥0 , such that {P(0)n} n≥0 equals the Determinant polynomial. We show that {P(k)n} n≥0 is VBP complete for k = 1 and it becomes VNP complete when k ≥ 2.In particular, P(k)n is Det =k n (X), a polynomial obtained by summing over all signed cycle covers that avoid length k cycles. We show that Det =1 n (X) is complete for VBP and Det =k n (X) is complete for VNP for all k ≥ 2 over any field F.3 Formally, VP is defined using topological approximations. However, for reasonably well-behaved fields F, the two notions of approximation are equivalent. We will focus on algebraic approximation in this note. 4 Let fn(x1, x2, . . . , x k(n) ) be a p-bounded polynomial family. fn is said to be in VNP if there exists a family gn ∈ VP such that fn =