# Optimal testing of multivariate polynomials over small prime fields

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## Abstract

We consider the problem of testing if a given function f:double-struck F q n → double-struck F q is close to a n-variate degree d polynomial over the finite field double-struck F q of q elements. The natural, low-query, test for this property would be to pick the smallest dimension t = t q, d, ≈ d/q such that every function of degree greater than dreveals this aspect on some t-dimensional affine subspace of double-struck F q n and to test that f when restricted to a random t-dimensional affine subspace is a polynomial of degree at most d on this subspace. Such a test makes only q t queries, independent of n. Previous works, by Alon et al. [1], and Kaufman and Ron [7] and Jutla et al. [6], showed that this natural test rejected functions that were Ω(1)-far from degree d-polynomials with probability at least Ω(q -t). (The initial work [1] considered only the case of q=2, while the work [6] only considered the case of prime q. The results in [7] hold for all fields.) Thus to get a constant probability of detecting functions that are at constant distance from the space of degree d polynomials, the tests made q 2t queries. Kaufman and Ron also noted that when q is prime, then q t queries are necessary. Thus these tests were off by at least a quadratic factor from known lower bounds. Bhattacharyya et al. [2] gave an optimal analysis of this test for the case of the binary field and showed that the natural test actually rejects functions that were Ω(1)-far from degree d-polynomials with probability Ω(1). In this work we extend this result for all fields showing that the natural test does indeed reject functions that are Ω(1)-far from degree d polynomials with Ω(1)-probability, where the constants depend only on q the field size. Thus our analysis thus shows that this test is optimal (matches known lower bounds) when q is prime. The main technical ingredient in our work is a tight analysis of the number of "hyper planes" (affine subspaces of co-dimension 1) on which the restriction of a degree dpolynomial has degree less than d. We show that the number of such hyper planes is at most O(q tq,d) - which is tight to within constant factors.

Original language English Proceedings - 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science, FOCS 2011 629-637 9 https://doi.org/10.1109/FOCS.2011.61 Published - 2011 Yes 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science, FOCS 2011 - Palm Springs, CA, United StatesDuration: 22 Oct 2011 → 25 Oct 2011

### Publication series

Name Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS 0272-5428

### Conference

Conference 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science, FOCS 2011 United States Palm Springs, CA 22/10/11 → 25/10/11

## Keywords

• Low-degree testing
• Property Testing
• Reed-Muller codes

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