Testing Polynomial Equivalence by Scaling Matrices

Abstract

In this paper we study the polynomial equivalence problem: test if two given polynomials f and g are equivalent under a non-singular linear transformation of variables.

We begin by showing that the more general problem of testing whether f can be obtained from g by an arbitrary (not necessarily invertible) linear transformation of the variables is equivalent to the existential theory over the reals. This strengthens an $\mathsf {NP}$-hardness result by Kayal [9].

Two n-variate polynomials f and g are said to be equivalent up to scaling if there are scalars $a_1, \ldots , a_n \in \mathbb {F} \setminus \{0\}$ such that $f(a_1x_1,\ldots , a_nx_n) = g(x_1,\ldots , x_n)$. Testing whether two polynomials are equivalent by scaling matrices is a special case of the polynomial equivalence problem and is harder than the polynomial identity testing problem.

As our main result, we obtain a randomized polynomial time algorithm for testing if two polynomials are equivalent up to a scaling of variables with black-box access to polynomials f and g over the real numbers.

An essential ingredient to our algorithm is a randomized polynomial time algorithm that given a polynomial as a black box obtains coefficients and degree vectors of a maximal set of monomials whose degree vectors are linearly independent. This algorithm might be of independent interest. It also works over finite fields, provided their size is large enough to perform polynomial interpolation.