On the structure of cubic and quartic polynomials

Abstract

In this paper we study the structure of polynomials of degree three and four that have high bias or high Gowers norm, over arbitrary prime fields. In particular we obtain the following results.1. Let f be a degree three polynomial with bias(f ) = δ then there exist r = O(log(1/δ)) quadratic polynomials. This result generalizes the corresponding result for quadratic polynomials. Let deg(f ) = 4 and biaswhere r = poly(1/δ), the i -s are linear, the q i -s are quadratics and the g i -s are cubic.3. Let deg(f ) = 4 and f U 4 = δ. Then there exists a partition of a subspaceItems 1,2 extend and improve previous results for degree three and four polynomials [KL08, GT07]. Item 3 gives a new result for the case of degree four polynomials with high U 4 norm. It is the first case where the inverse conjecture for the Gowers norm fails [LMS08,GT07], namely that such an f is not necessarily correlated with a cubic polynomial. Our result shows that instead f equals a cubic polynomial on a large subspace (in fact we show that a much stronger claim holds).Our techniques are based on finding a structure in the space of partial derivatives of f . For example, when deg(f ) = 4 and f has high U 4 norm we show that there exist quadratic polynomials {q i } i∈ [r] and linear functions { i } i∈ [R] such that (on a large enough subspace) every partial derivative of f can be written as ∆ y (f ) =

Fourier Sparsity, Spectral Norm, and the Log-Rank Conjecture

“…In [HS10], Haramaty and Shpilka proved that rank(f ) = O(log 2 (1/ f U 3 )) = O(log 2 (1/bias(f ))) for degree-3 polynomials. For degree-4 polynomials, however, the bound gets exponentially worse, and there were no results for higher degrees.…”

“…The following notion of polynomial rank has been studied in [Dic58] for degree-2 polynomials and in [HS10] for degree-3 polynomials. 4 Definition 2.…”

“…We prove Theorem 5 for the special case of cubic polynomials first. This is because degree 3 is the first non-trivial case and we use this result in our final induction proof of Theorem 5; more importantly, the proof applies some ideas from [HS10] which inspire our proof for the general constant degree case.…”

Additive Combinatorics: With a View Towards Computer Science and Cryptography—An Exposition

“…To the best of my knowledge, there are two types of inverse theorems in additive combinatorics, namely the inverse sumset theorems of Freiman type (see, e.g., [78,117,118,152,155,275,299,306,307,308,318] and [116,238]), and inverse theorems for the Gowers norms (see, e.g., [143,146,147,148,153,162,156,159,163,164,176,181,189,203,224,225,233,270,326,332]). It is interesting that the inverse conjecture leads to a finite field version of Szemerédi's theorem [320]: Let F p be a finite field.…”

“…Additive combinatorics has recently found a great deal of remarkable applications to computer science and cryptography; for example, to expanders [20,21,38,44,52,53,54,55,62,63,66,105,106,167,206,283,342], extractors [19,20,26,28,38,41,103,104,107,167,182,217,346,350], pseudorandomness [33,223,226,227,301] (also, [331,334] are two surveys and [335] is a monograph on pseudorandomness), property testing [31,176,177,181,203,204,270,281,332] (see also …”