On Hardness of Testing Equivalence to Sparse Polynomials Under Shifts

Suryajith Chillara*, Coral Grichener*, Amir Shpilka*

*Corresponding author for this work

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review


We say that two given polynomials f, g ∈ R[x1, ..., xn], over a ring R, are equivalent under shifts if there exists a vector (a1, ..., an) ∈ Rn such that f(x1 + a1, ..., xn + an) = g(x1, ..., xn). 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 SparseShiftR 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 SparseShiftR is at least as hard as checking if a given system of polynomial equations over R[x1, ..., xn] has a solution (Hilbert’s Nullstellensatz). As a consequence of this reduction, we get the following results. 1. SparseShiftZ 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, SparseShiftR is also NPR-complete. In particular, SparseShiftZ is also NPZ-complete. We also study the gap version of the SparseShiftR and show the following. 1. For every function β : N → R+ such that β ∈ o(1), Nβ-gap-SparseShiftZ 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-SparseShiftR 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 ≤ nO(1) in the arithmetic circuit model, the αd-gap-SparseShiftR problem is NP-hard when given polynomials of degree at most d, in O(nd) many variables, as input.

Original languageEnglish
Title of host publication40th International Symposium on Theoretical Aspects of Computer Science, STACS 2023
EditorsPetra Berenbrink, Patricia Bouyer, Anuj Dawar, Mamadou Moustapha Kante
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
ISBN (Electronic)9783959772662
StatePublished - 1 Mar 2023
Event40th International Symposium on Theoretical Aspects of Computer Science, STACS 2023 - Hamburg, Germany
Duration: 7 Mar 20239 Mar 2023

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
ISSN (Print)1868-8969


Conference40th International Symposium on Theoretical Aspects of Computer Science, STACS 2023


FundersFunder number
Blavatnik Family Foundation
Israel Science Foundation514/20


    • Hilbert’s Nullstellensatz
    • algebraic complexity
    • hardness of approximation
    • polynomial equivalence
    • shift equivalence


    Dive into the research topics of 'On Hardness of Testing Equivalence to Sparse Polynomials Under Shifts'. Together they form a unique fingerprint.

    Cite this