#### Title

Certain diagonal equations over finite fields

2009

Thesis

M.A.

#### Degree Granting Department

Mathematics and Statistics

#### Major Professor

Xiang-Dong Hou, Ph.D.

#### Keywords

Irreducible polynomial, Gaussian sum, Planar function, Hasse-Weil bound, Elliptic curve

#### Abstract

Let F q to the t be the finite field with q to the t elements and let F q to the t star be its multiplicative group. We study the diagonal equation a times x to the (q minus 1) plus b times y to the (q minus 1) equals c, where a,b and c are elements of F q to the t star. This equation can be written as x to the (q minus 1) plus alpha times y to the (q minus 1) equals beta, where alpha and beta are elements of F q to the t star. Let N sub t (alpha,beta) denote the number of solutions (x,y) in F q to the t star cross F q to the t star of the equation x to the (q minus 1) plus alpha times y to the (q minus 1) equals beta and I(r;a,b) be the number of monic irreducible polynomials f with coefficients in F q of degree r with f(0) equals a and f(1) equals b. We show that N sub t (alpha,beta) can be expressed in terms of I(r;a,b), where r divides t and a,b are elements of F q star are related to alpha and beta. A recursive formula for I(r;a,b) will be given and we illustrate this by computing I(r;a,b) for r greater than or equal to 2 but less than or equal to 4. We also show that N sub 3 (alpha,beta) can be expressed in terms of the number of monic irreducible cubic polynomials over F q with prescribed trace and norm. Connsequently, N sub 3 (alpha,beta) can be expressed in terms of the number of rational points on a certain elliptic curve. We give a proof that given any a,b elements of F q star and integer r greater than or equal to 3, there always exists a monic irreducible polynomial f with coefficients in F q of degree r such that f(0) equals a and f(1) equals b. We also use the result on N sub 2 (alpha,beta) to construct a new family of planar functions.

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