On Hardness of Testing Equivalence to Sparse Polynomials Under Shifts

We say that two given polynomials f, g ∈ R[x₁, ..., x_n], over a ring R, are equivalent under shifts if there exists a vector (a₁, ..., a_n) ∈ Rⁿ such that f(x₁ + a₁, ..., x_n + a_n) = g(x₁, ..., x_n). This is a special variant of the polynomial projection problem in Algebraic Complexity Theory. Grigoriev and Karpinski (FOCS 1990), Lakshman and Saunders (SIAM J. Computing, 1995), and Grigoriev and Lakshman (ISSAC 1995) studied the problem of testing polynomial equivalence of a given polynomial to any t-sparse polynomial, over the rational numbers, and gave exponential time algorithms. In this paper, we provide hardness results for this problem. Formally, for a ring R, let SparseShift_R be the following decision problem – Given a polynomial P(X), is there a vector a such that P(X + a) contains fewer monomials than P(X). We show that SparseShift_R is at least as hard as checking if a given system of polynomial equations over R[x₁, ..., x_n] has a solution (Hilbert’s Nullstellensatz). As a consequence of this reduction, we get the following results. 1. SparseShift_Z is undecidable. 2. For any ring R (which is not a field) such that HNR is NPR-complete over the Blum-Shub-Smale model of computation, SparseShift_R is also NPR-complete. In particular, SparseShift_Z is also NP_Z-complete. We also study the gap version of the SparseShift_R and show the following. 1. For every function β : N → R+ such that β ∈ o(1), N^β-gap-SparseShift_Z is also undecidable (where N is the input length). 2. For R = Fp, Q, R or Zq and for every β > 1 the β-gap-SparseShift_R problem is NP-hard. Furthermore, there exists a constant α > 1 such that for every d = O(1) in the sparse representation model, and for every d ≤ n^O(1) in the arithmetic circuit model, the α^d-gap-SparseShift_R problem is NP-hard when given polynomials of degree at most d, in O(nd) many variables, as input.