Testing low‐degree polynomials over prime fields

Abstract

ABSTRACT:We present an efficient randomized algorithm to test if a given function f : F n p → F p (where p is a prime) is a low-degree polynomial. This gives a local test for Generalized Reed-Muller codes over prime fields. For a given integer t and a given real > 0, the algorithm queries f at O(p−1 +1 ) points to determine whether f can be described by a polynomial of degree at most t. If f is indeed a polynomial of degree at most t, our algorithm always accepts, and if f has a relative distance at least from every degree t polynomial, then our algorithm rejects f with probability at least 1 2 . Our result is almost optimal since any such algorithm must query f on at least ( 1 + p t+1 p−1 ) points.

Optimal Testing of Multivariate Polynomials over Small Prime Fields

“…Finally they analyzed this O(q t ) query test, showing that the δ-soundness of this test is Ω(δq −t ), again leading to an absolutely sound test with query complexity O(q 2t ) which is off by a quadratic factor. The proof techniques of [AKK + 05] and [KR06,JPRZ04] were similar and indeed the subsequent generalization of Kaufman and Sudan [KS08] shows how these results fall in the very general framework of "affine-invariant" property testing, where again all known tests are off by (at least) a quadratic factor.…”

“…The setting of general q was considered by Kaufman and Ron [KR06] and independently (for the case of prime q) by Jutla et al [JPRZ04]. They (in particular [KR06]) showed that there exists an integer t = t q,d ≈ d/q (we will be more precise with this later) such that the natural test for low-degreeness makes Ω(q t ) queries.…”

“…Such a test makes only q t queries, independent of n. Previous works, by Alon et al [AKK + 05], and Kaufman and Ron [KR06] and Jutla et al [JPRZ04], showed that this natural test rejected functions that were Ω(1)-far from degree d-polynomials with probability at least Ω(q −t ) (the results of [KR06] hold for all fields F q , while the results of [JPRZ04] hold only for fields of prime order). Thus to get a constant probability of detecting functions that were at constant distance from the space of degree d polynomials, the tests made q 2t queries.…”

Sublinear time algorithms

“…A long series of works have given increasingly robust characterizations of functions that are low total degree polynomials (cf. [6], [32], [7], [34], [3], [29], [27]). …”

Symmetric Functions Capture General Functions

“…(ii) Under certain restrictions on the degree of f and |F|, we can talk about worst-case complexity of f and f sym . In particular, we use well-known properties of Reed-Muller codes that have been used numerous times in the local testing algorithms and PCP constructions [3,2,1,16,15]. However, unlike the local testing results which need to handle errors, in our application we only need to handle erasures-roughly because we can efficiently determine the degree of F without computing the vector a, which leads to better bounds.…”