NP-hardness of testing equivalence to sparse polynomials and to constant-support polynomials

Abstract

Given a list of monomials of a n-variate polynomial f over a field F, and an integer s, decide whether there exists an invertible transform A and a b such that f(Ax + b) has less than s monomials. This problem is called the Equivalence testing to sparse polynomials (ETsparse). It was studied in [GrigorievK93] over Q, in this work, they give an exponential in n^4 time algorithm for the problem. The lack of progress in the complexity of the problem over last three decades raises a question, is ETsparse hard? In this thesis we give an affirmative answer to the question by showing that it is NP-hard over any field.

Sparse orbit complexity of a polynomial f is the smallest integer s_0 such that there exists an invertible transform A such that f(Ax) has s_0 monomials. Since ETsparse is NP-hard hence computing the sparse orbit complexity is also NP-hard. We also show that approximating the sparse orbit complexity upto a factor of s_f^{1/3-\epsilon} for any \epsilon \in (0,1/3) is NP-hard, where s_f is the number of monomials in f. Interestingly, this approximation result has been shown without invoking the celebrated PCP theorem.

[ChillaraGS23] study a variant of the problem which focus on shift equivalence. More precisely, given f over some ring R (the input has the same representation as in ETsparse) and an integer s, does there exists a b such that f(x + b) has less than s monomials. It is called the SETsparse problem, [ChillaraGS23] showed that SETsparse is NP-hard when R is an integral domain which is not a field; we extend their result to the case when R is a field.

Finally, we also study the problem of testing equivalence to constant-support polynomials; more precisely, given a polynomial f as before and with support \sigma, does there exists an invertible transform A such that f(Ax) has support \sigma -1. We call this problem ETsupport. We show that ETsupport is NP-hard for \sigma >= 5 and over any field.