A Besicovitch set in AG(n, q) is a set of points containing a line in every direction. The Kakeya problem is to determine the minimal size of such a set. We solve the Kakeya problem in the plane, and substantially improve the known bounds for n > 4.

For integers, the parameter n is the bit-length, and for the finite field Fq we let n = log q. In the case of polynomial root-finding, d is the degree of the polynomial and we list bounds on the expected running time since these operations are most efficiently implemented using probabilistic algorithms. In Lecture 3 we addressed the cost of addition and subtraction in both Z and Fq, and the cos...

Non-linear polynomial systems over finite fields are used to model functional behavior of cryptosystems, with applications in system security, computer cryptography, and post- quantum cryptography. Solving is also one the most difficult problems mathematics. In this paper, we propose an automated reasoning procedure for deciding satisfiability a non-linear equations fields. We introduce zero de...