Linear Projections of the Vandermonde Polynomial

TL;DR

This paper investigates the structural and computational properties of linear projections of Vandermonde polynomials, providing algorithms for equivalence testing and analyzing their symmetry groups.

Contribution

It introduces a deterministic polynomial-time algorithm for testing linear equivalence of Vandermonde polynomials given as product of linear factors, and explores their symmetry structures.

Findings

Deterministic polynomial-time algorithm for equivalence testing with explicit factorization.

Randomized polynomial-time algorithm for black-box polynomial equivalence testing.

Identification of the simple Lie algebra associated with Vandermonde polynomial symmetries.

Abstract

An n-variate Vandermonde polynomial is the determinant of the n x n matrix where the ith column is the vector (1, x_i, x_i^2, ...., x_i^{n-1})^T. Vandermonde polynomials play a crucial role in the theory of alternating polynomials and occur in Lagrangian polynomial interpolation as well as in the theory of error correcting codes. In this work we study structural and computational aspects of linear projections of Vandermonde polynomials. Firstly, we consider the problem of testing if a given polynomial is linearly equivalent to the Vandermonde polynomial. We obtain a deterministic polynomial time algorithm to test if the polynomial f is linearly equivalent to the Vandermonde polynomial when f is given as product of linear factors. In the case when the polynomial f is given as a black-box our algorithm runs in randomized polynomial time. Exploring the structure of projections of Vandermonde polynomials further, we describe the group of symmetries of a Vandermonde polynomial and show that the associated Lie algebra is simple.