Variants of the Determinant Polynomial and the $$\textsf {}$$-Completeness

The determinant is a canonical \(\textsf {VBP}\)-complete polynomial in the algebraic complexity setting. In this work, we introduce two variants of the determinant polynomial which we call \(\mathtt{StackDet}_n(X)\) and \(\mathtt{CountDet}_n(X)\) and show that they are \(\textsf {VP}\) and \(\textsf {VNP}\) complete respectively under p-projections. The definitions of the polynomials are inspired by a combinatorial characterisation of the determinant developed by Mahajan and Vinay (SODA 1997). We extend the combinatorial object in their work, namely clow sequences, by introducing additional edge labels on the edges of the underlying graph. The idea of using edge labels is inspired by the work of Mengel (MFCS 2013).