Representation of Boolean Functions by Polynomials

Student: Kryukova Ekaterina

Supervisor: Vladimir V. Podolskii

Educational Programme: Applied Mathematics and Information Science (Bachelor)

Year of Graduation: 2020

We consider the Lin-rank Conjecture, which states that if many linear constraints are necessary to reduce the degree of a polynomial, then the Fourier sparsity will be large. Using the observation that the linear rank depends only on maximum degree monoms, we study the methods proposed in the Fourier Sparsity of GF(2) Polynomials. Namely, methods for proving lower bounds on Fourier sparsity for two classes of functions: with a complete set of maximum degree monomes and with pairwise disjoint maximum degree monomes. We apply these methods to six new classes of functions, and also prove upper bounds on the linear rank value for these classes of functions. In a number of cases, we prove that this estimate is an exact value of linear rank.

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