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

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