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 Detn≠k(X), a polynomial obtained by summing over all signed cycle covers that avoid length k cycles. We show that Detn≠1(X) is complete for VBP and Detn≠k(X) is complete for VNP for all k≥ 2 over any field F.