## Abstract

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}) ∈ R^{n} 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 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.

Original language | English |
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Title of host publication | 40th International Symposium on Theoretical Aspects of Computer Science, STACS 2023 |

Editors | Petra Berenbrink, Patricia Bouyer, Anuj Dawar, Mamadou Moustapha Kante |

Publisher | Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing |

ISBN (Electronic) | 9783959772662 |

DOIs | |

State | Published - 1 Mar 2023 |

Event | 40th International Symposium on Theoretical Aspects of Computer Science, STACS 2023 - Hamburg, Germany Duration: 7 Mar 2023 → 9 Mar 2023 |

### Publication series

Name | Leibniz International Proceedings in Informatics, LIPIcs |
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Volume | 254 |

ISSN (Print) | 1868-8969 |

### Conference

Conference | 40th International Symposium on Theoretical Aspects of Computer Science, STACS 2023 |
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Country/Territory | Germany |

City | Hamburg |

Period | 7/03/23 → 9/03/23 |

### Funding

Funders | Funder number |
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Blavatnik Family Foundation | |

Israel Science Foundation | 514/20 |

## Keywords

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