On The Closures of Monotone Algebraic Classes and Variants of the Determinant

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 =