Polynomials and harmonic functions on finite fields

Abstract

__The aim of this work is to see under what circumstances it is possible to extend the generalized form of the Lagrange Interpolation Formula which states that there is a unique polynomial f of degree n-1 which satisfies the condition: if x1, [see PAI] are points of the real or complex plane respectively, where ,[see PAI] • with multiplicity m1, [ ] ,a are real or complex numbers, then [] [] __In Chapter I this result is extended to the finite fields. Here we work with polynomials as in the classical case. To start, new "derivatives II are defined on the finite field polynomials, and are shown to have properties which can determine the multiplicity of the roots of a polynomial as the ordinary derivative determines multiplicity in the real and complex cases. Using these new operators we prove that a generalized form of the Lagrange Interpolation Formula holds in the finite fields. __Still working in the finite field we try to extend our_ result to complex valued functions. Chapter II deals with the characters as replacements for the xn. We hope that the characters will share with the classical polynomials the property of being harmonic, so we start by looking for a definition of harmonic. The first definition we give is a mean value property where we sum over the units . What seems to be an obvious choice for a definition is shown to be unsatisfactory when we prove that only constant functions are harmonic by this definition. A second definition in which we sum over a smaller group is also tried with the same results. Our final definition alters the first by limiting the group we sum over and the set of radii. Using this definition we show that non-constant, harmonic, complex valued functions do exist. We also show that if the finite field is GF(p n) and the set of radii contains n appropriately chosen elements, the only harmonic functions are the constant functions. Corresponding definitions and results are given for Q, the field of p-adic , and for the field of formal power series. As a note of comparison of these results with the classical case we give a statement of the Two Radius Theorem. __Since the set of radii easily becomes large enough to leave us with only the constant functions for our harmonic functions, it does not seem worthwhile to pursue the subject of harmonic functions any further. Instead, we turn to looking for "derivatives" on the characters to use in defining multiplicity in hopes we can use them in proving an interpolation formula. In the case of the multiplicative characters we show no such operators exist. __Although we have interesting results on harmonicity, none encourages us to use the characters to replace the x . Therefore, we continue in Chapter III to look for some other complex valued functions on the finite field which we can use. The candidates in Chapter III are Legendre polynomials. After looking at the classical Legendre polynomials we give a corresponding definition for Legendre polynomials and associated Legendre polynomials on the finite field. Next we actually compute examples of these polynomials in GF(3), GF(S), GF(9), GF(25) and GF(7). We meet with only limited success in making a correspondence between the new polynomials and the classical ones.