Gabber’s presentation lemma for finite fields

Gauss's Lemma for Number Fields

A Removal Lemma for Systems of Linear Equations over Finite Fields

We prove a removal lemma for systems of linear equations over finite fields: let X1, . . . , Xm be subsets of the finite field Fq and let A be a (k×m) matrix with coefficients in Fq and rank k; if the linear system Ax = b has o(q) solutions with xi ∈ Xi, then we can destroy all these solutions by deleting o(q) elements from each Xi. This extends a result of Green [Geometric and Functional Analy...

Expanding polynomials over finite fields of large characteristic, and a regularity lemma for definable sets

Let P : F × F → F be a polynomial of bounded degree over a finite field F of large characteristic. In this paper we establish the following dichotomy: either P is a moderate asymmetric expander in the sense that |P (A,B)| |F| whenever A,B ⊂ F are such that |A||B| ≥ C|F|2−1/8 for a sufficiently large C, or else P takes the form P (x, y) = Q(F (x) + G(y)) or P (x, y) = Q(F (x)G(y)) for some polyn...

Classical Wavelet Transforms over Finite Fields

This article introduces a systematic study for computational aspects of classical wavelet transforms over finite fields using tools from computational harmonic analysis and also theoretical linear algebra. We present a concrete formulation for the Frobenius norm of the classical wavelet transforms over finite fields. It is shown that each vector defined over a finite field can be represented as...

Classical wavelet systems over finite fields

This article presents an analytic approach to study admissibility conditions related to classical full wavelet systems over finite fields using tools from computational harmonic analysis and theoretical linear algebra. It is shown that for a large class of non-zero window signals (wavelets), the generated classical full wavelet systems constitute a frame whose canonical dual are classical full ...