# Polynomials and harmonic functions on finite fields

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## 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.