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 \(C \in \{\textsf {mVF}, \textsf {mVP}, \textsf {mVNP}\}\), \(C = \overline{C}\). Here, \(\textsf {mVF}, \textsf {mVP}\), and \(\textsf {mVNP}\) are monotone variants of \(\textsf {VF}, \textsf {VP}\), and \(\textsf {VNP}\), respectively. For an algebraic complexity class C, \(\overline{C}\) denotes the closure of C. For \(\textsf {mVBP}\) a similar result was shown in Bläser et al. (in: 35th Computational Complexity Conference, CCC 2020. LIPIcs, vol 169, pp 21–12124, 2020. https://doi.org/10.4230/LIPIcs.CCC.2020.21). Here we extend their result by adapting their proof.

• We define polynomial families \(\{\mathcal {P}(k)_n\}_{n \ge 0}\), such that \(\{\mathcal {P}(0)_n\}_{n \ge 0}\) equals the determinant polynomial. We show that \(\{\mathcal {P}(k)_n\}_{n \ge 0}\) is \(\textsf {VBP}\) complete for \(k=1\) and it becomes \(\textsf {VNP}\) complete when \(k \ge 2\). In particular, \(\{\mathcal {P}(k)_n\}\) is \(\mathtt {Det^{\ne k}_n(X)}\), a polynomial obtained by summing over all signed cycle covers that avoid length k cycles. We show that \(\mathtt {Det^{\ne 1}_n(X)}\) is complete for \(\textsf {VBP}\) and \(\mathtt {Det^{\ne k}_n(X)}\) is complete for \(\textsf {VNP}\) for all \(k \ge 2\) over any field \(\mathbb {F}\).