Solutions to Systems of Equations over Finite Fields

This dissertation investigates the existence of solutions to equations over finite fields with an emphasis on diagonal equations. In particular: Given a system of equations, how many solutions are there? In the case of a system of diagonal forms, when does a nontrivial solution exist? Many results are known that address (1) and (2), such as the classical Chevalley--Warning theorems. With respect to (1), we have improved a recent result of D.R. Heath--Brown, which provides a lower bound on the total number of solutions to a system of polynomials equations. Furthermore, we have demonstrated that several of our lower bounds are sharp under the stated hypotheses. With respect to (2), we have several improvements that extend known results. First, we have improved a result of James Gray by extending his theorem to a larger class of equations. Second, for particular degrees, number of forms, and finite fields, we have determined the minimal number of variables needed to guarantee the existence of a nontrivial solution. Third, there are many results, which address (2) for particular types of systems known as A-systems. We give a criterion that characterizes when a system of equations is an A-system. Finally, we have provided exposition that adds significantly more detail to two important papers by Tietäväinen

On Solving Systems of Diagonal Polynomial Equations Over Finite Fields

We present an algorithm to solve a system of diagonal polynomial equations over finite fields when the number of variables is greater than some fixed polynomial of the number of equations whose degree depends only on the degree of the polynomial equations. Our algorithm works in time polynomial in the number of equations and the logarithm of the size of the field, whenever the degree of the polynomial equations is constant. As a consequence we design polynomial time quantum algorithms for two algebraic hidden structure problems: for the hidden subgroup problem in certain semidirect product p-groups of constant nilpotency class, and for the multi-dimensional univariate hidden polynomial graph problem when the degree of the polynomials is constant.Comment: A preliminary extended abstract of this paper has appeared in Proceedings of FAW 2015, Springer LNCS vol. 9130, pp. 125-137 (2015

Graham Higman's PORC theorem

Graham Higman published two important papers in 1960. In the first of these papers he proved that for any positive integer $n$ the number of groups of order $p^{n}$ is bounded by a polynomial in $p$, and he formulated his famous PORC conjecture about the form of the function $f(p^{n})$ giving the number of groups of order $p^{n}$. In the second of these two papers he proved that the function giving the number of $p$-class two groups of order $p^{n}$ is PORC. He established this result as a corollary to a very general result about vector spaces acted on by the general linear group. This theorem takes over a page to state, and is so general that it is hard to see what is going on. Higman's proof of this general theorem contains several new ideas and is quite hard to follow. However in the last few years several authors have developed and implemented algorithms for computing Higman's PORC formulae in special cases of his general theorem. These algorithms give perspective on what are the key points in Higman's proof, and also simplify parts of the proof. In this note I give a proof of Higman's general theorem written in the light of these recent developments

Non-extremal black holes from the generalised r-map

We review the timelike dimensional reduction of a class of five-dimensional theories that generalises 5D, N = 2 supergravity coupled to vector multiplets. As an application we construct instanton solutions to the four-dimensional Euclidean theory, and investigate the criteria for solutions to lift to static non-extremal black holes in five dimensions. We focus specifically on two classes of models: STU-like models, and models with a block diagonal target space metric. For STU-like models the second order equations of motion of the four-dimensional theory can be solved explicitly, and we obtain the general solution. For block diagonal models we find a restricted class of solutions, where the number of independent scalar fields depends on the number of blocks. When lifting these solutions to five dimensions we show, by explicit calculation, that one obtains static non-extremal black holes with scalar fields that take finite values on the horizon only if the number of integration constants reduces by exactly half.Comment: 22 pages. Based on talk by OV at "Black Objects in Supergravity School" (BOSS2011), INFN, Frascati, Italy, 9-13 May, 201