On Hardness of Testing Equivalence to Sparse Polynomials Under Shifts
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We say that two given polynomials f, g ∈ R[X], over a ring R, are equivalent under shifts if there exists a vector a ∈ R^n such that f(X+a) = g(X). Grigoriev and Karpinski (FOCS 1990), Lakshman and Saunders (SICOMP, 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[x_1,…, 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 HN_R is NP_R-complete over the Blum-Shub-Smale model of computation, SparseShift_R is also NP_R-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=𝔽_p, ℚ, ℝ or ℤ_q and for every β>1 the β-gap-SparseShift_R problem is NP-hard.
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