On Hardness of Testing Equivalence to Sparse Polynomials Under Shifts

Authors Suryajith Chillara, Coral Grichener, Amir Shpilka

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Suryajith Chillara
  • International Institute of Information Technology, Hyderabad, India
Coral Grichener
  • Google Research, Herzliya, Israel
Amir Shpilka
  • Tel Aviv University, Israel

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Suryajith Chillara, Coral Grichener, and Amir Shpilka. On Hardness of Testing Equivalence to Sparse Polynomials Under Shifts. In 40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 254, pp. 22:1-22:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


We say that two given polynomials f, g ∈ R[x_1, ..., x_n], over a ring R, are equivalent under shifts if there exists a vector (a_1, ..., a_n) ∈ Rⁿ such that f(x_1+a_1, ..., x_n+a_n) = g(x_1, ..., 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[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_ 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. 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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Circuit complexity
  • Theory of computation → Algebraic complexity theory
  • algebraic complexity
  • shift equivalence
  • polynomial equivalence
  • Hilbert’s Nullstellensatz
  • hardness of approximation


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