**Note:** The versions of these papers that I have put here are preprints. They
may differ from the final published version.

**In Preparation**

[1] *Non-homogeneous diagonal equations*.

[2] (with C. Broll, J. Jennings, P.H.A. Rodrigues and D. Veras) *More exact values of the function Gamma*(k)*.

[3] (with H. Godinho and B. Miranda) *Sextic forms over Q_2(\sqrt{5})*.

**Submitted**

[4]

__Abstract:__ Let *a* be an integer greater than 2, and define a sequence recursively by
*G _{0}=a*,

**Accepted for Publication / In Print**

[5]

__Abstract:__ In this article, we answer the following question about graph pebbling. Consider a cycle graph with
*n* vertices and all edges directed counterclockwise, and choose a positive integer *d* with *d* <
*n*. We determine the smallest number *s* such that if *s* pebbles are placed on the graph,
then there is at least one pebble which can be moved a distance of at least *d* vertices away from its original
position through a sequence of pebbling moves.

[6] (with H. Godinho)

__Abstract:__ The number *Γ(k)* is defined as the least integer *s* such that the diagonal form
*a _{1}x_{1}^{k} + ... +
a_{s}x_{s}^{k}* is guaranteed to have a nontrivial zero in every

[7] (with H. Godinho, P. H. A. Rodrigues, and D. Veras)

__Abstract:__ In this article, we calculate an upper bound for the value of *Gamma^*(k,p)*, which gives the
smallest number of variables necessary to guarantee that any additive form of degree *k* with integer coefficients
has a *p*-adic solution, and also give a condition under which this upper bound is actually an equality.
Additionally, we calculate the value of *Gamma^*(54)*, which is the smallest number of
variables needed to guarantee that any additive form of degree 54 with integer coefficients has a zero in every
*p*-adic field **Q**_{p}.

[8]

__Abstract:__ Consider an additive form * a _{1}x_{1}^{6} + ... +
a_{s}x_{s}^{6}*
whose coefficients are integers of one of the six unramified quadratic extensions of the

[9]

__Abstract:__ Consider an additive form * a _{1}x_{1}^{d} + ... +
a_{s}x_{s}^{d}*
whose coefficients are all 2-adic integers. In this article we give an exact formula, in terms of

[10] (With R. Auer)

__Abstract:__ Major League Baseball's decision more than a decade ago to add wild card teams to
its playoff structure remains controversial even now. In this article, we attempt to support the
notion that there should be wild card teams. To do this, we make a very simple probabalistic model
of Major League Baseball's final regular-season standings. Then we use this model to predict how often
the team that becomes the wild card will have a better record than the worst division-winning team, and
calculate the expected rank in the league of the worst division winner. Perhaps surprisingly, our model
predicts that the worst division winner should finish 7th or worse in the league more than 12% of the
time.

[11] (With A. Fox)

__Abstract:__ In this article, we treat sequences formed by the same procedure as the ones
in the article "Weakly complete sequences formed by the greedy algorithm" below. We show that the
patterns described in that abstract will always exist whenever 1* < d < n-*7. This majority of this
work was done by Alyson Fox while she was my student in the Hauber summer research program at Loyola.

[12] (With H. Godinho and P. H. A. Rodrigues)

__Abstract:__ This article shows that a system of two homogeneous additive polynomials of degree 6
with integer coefficients will have a nontrivial *p*-adic solution for any prime *p*
provided only that the system has a total of 73 variables.

[13]

__Abstract:__ Suppose that *k* and *n* are odd
integers. We show that any system of diagonal
forms of degrees *k* and *n*, with integer coefficients, in
at least *k ^{2}+n^{2}+*1 variables has a nontrivial
zero in the field

[14] (With M. Paul)

__Abstract:__ Consider a sequence defined recursively by the following procedure. Pick numbers
*n* and *d*, and set *a*_{1}*=n* and *a*_{2}*=n+d*.
Then for each *k>*2, set *a _{k}* to be the smallest number greater than

[15]

__Abstract:__ The function *Gamma ^{*}(k)* gives the smallest number of variables
required to ensure that an additive form of degree

[16] (With K. Mohamed and Z. Tahar)

__Abstract:__ Suppose that *a* is a *p*-adic number. In this paper, we
consider the problem of calculating the *p*-adic digits of the square root of *a*.
We do this using two fixed-point methods with quadratic and cubic rates of convergence. For
each method, we calculate the minimum number of correct digits obtained after *n*
iterations, and the number of iterations required to obtain an approximation of any given
accuracy.

[17] (With C. Xenophontos)

__Abstract:__ In this expository paper, we show how rootfinding methods from
numerical analysis (Newton's method and the secant method) can be used to
calculate inverses of numbers modulo powers of primes. In undergraduate courses,
the fields of numerical analysis and number theory can appear to be unrelated, and
we believe that students may enjoy seeing a connection between them.

[18]

__Abstract:__ In this expository paper, we derive a formula for the sum of a series of
sines or cosines, where the angles form an arithmetic progression. This derivation of the
formula was originally given in the journal Arbelos, which is now sadly out of print. The
derivation is interesting because it does not make any use of complex numbers.

[19]

__Abstract:__ This paper is similar to [3] below, except that we are able to remove the
restriction that the degrees of the forms must be all different. Given a system of diagonal
forms over **Q**_{p}, we ask how many variables are required to guarantee that the
system has a nontrivial zero. We show that if the prime p satisfies
p > (largest degree) - (smallest degree) + 1,
then there is a bound on the sufficient number of variables which is a polynomial in the degrees
of the forms. A result of Lewis & Montgomery implies that any bound which works for all primes
must exhibit exponential growth. So this result can be thought of as bounding how small p can be
before exponential growth is required.

[20]

(

__Abstract:__ In 1966, Davenport & Lewis published their paper *Notes on
congruences III*, in which they proved that under some mild conditions, a system of
two additive forms must have a nonsingular simultaneous zero modulo any prime number. In
their paper, they ask whether the theorem is true in general finite fields, and point
out that one of their key lemmas is no longer true in this situation. In this paper we
answer their question in the affirmative, proving that under the same conditions, a system
of two additive forms over a finite field must have a nonsingular simultaneous zero. We then
apply this result to obtain an upper bound on the number of variables required to ensure that
a system of two additive forms of equal degrees has a nontrivial zero in a *p*-adic field.

[21]

__Abstract:__ It is known that any system of diagonal forms has a
nontrivial *p*-adic zero for all primes *p* provided that
the number of variables is large enough in terms of the degrees. A result
of Lewis & Montgomery shows that the required number of variables must
exhibit exponential growth. However, a theorem of Ax & Kochen states that
if *p* is large enough, then a bound which exhibits only polynomial
growth suffices. The purpose of this paper is to bound the primes for
which exponential growth is required in the situation where the degrees
are all different. In particular, if the degrees of the forms are
*k*_{1}* > ... >k*_{R} and if *p >
k*_{1}*-k*_{R}*+*1, then we give a
bound
for
the required
number of variables which has only polynomial growth. This bound is
larger than the Ax-Kochen bound, but we note that it applies for primes
which are smaller than the largest of the degrees.

[22]

__Abstract:__ In this paper we develop a bound on the number of
variables required to guarantee that two diagonal homogeneous polynomials
of different degrees *k* and *n* with coefficients in
* Q_{p}* have a nontrivial simultaneous zero in

[23]

__Abstract:__ A recent paper in Math Magazine gave a proof that there are exactly
*p ^{2}(p^{2}+2p-1)/2* two-by-two matrices with entries in

[24]

__Abstract:__ It is known by work of Ax & Kochen that given any
natural number *d*, any homogeneous polynomial of degree *d*
in *d*^{2}+1 variables with coefficients in
* Q_{p}* has a nontrivial zero provided that

[25]

__Abstract:__ The main result of this paper is that any system of
*R* diagonal (additive) homogeneous polynomials of degree
*k* in at least 4*R*^{2}*k*^{2}
variables with coefficients in * Q_{p}* has a
nontrivial simultaneous zero in

[26] (With P. Isihara)

__Abstract:__ In this expository paper, we introduce the reader to the analysis
of musical chords using the additive group **Z**/12**Z** and the twelve-tone
operators, which can be thought of as functions on **Z**/12**Z**.

**Ph.D. Thesis**

[24] *Forms in many variables over p-adic
fields.* Ph.D. thesis, University of Michigan, 2000.
[PDF]
[PS]
[DVI]

__Abstract:__ This is my Ph.D. thesis. The content includes that of
papers [4] and [5] above, and also some of [1]. Additionally, we
elaborate on work of Skinner to give a bound on the number of variables
necessary to guarantee that there exists a nontrivial simultaneous zero of
a system of *R* homogeneous additive polynomials of degree
*p*^{t} with coefficients in a finite extension of
* Q_{p}*. This bound does not depend on the degree
of the field extension. There is also an introduction to the field of
study of zeros of homogeneous equations over