Michael P. Knapp

I have attempted to write this description on a level that can be understood by an undergraduate student who has had two semesters of calculus. I hope that I have succeeded. If you believe that something is written poorly, please feel free to email me and offer any suggestions for improvement. If you are looking for a more advanced account of my research, please read the grant proposal that I submitted to the National Science Foundation in October 2001 (the references cited in the proposal are here).

I think that this is very readable as an html document, but that the
formatting is a bit nicer as a pdf file. If you'd rather read that, then
just click here.

My research is in the field of number theory, and is closely related
to the problem of determining whether an equation or system of equations
has a solution in which all of the variables are integers. This has been
considered an interesting problem in math for a long time, at least since
the time of the ancient Greeks, who considered a problem insoluble unless
there was a solution in integers. For example, the Greeks would have said
that the equation 2*x*-1=0 has no solutions because there is no __integer__
that you can substitute for *x* which makes the equation true. Unfortunately,
it can be proven that
there is no procedure that you can always follow no matter what equation (or
system of equations) you are given which can tell whether the equation
has an integral solution. In fact, this is even
true if we only care about polynomial equations. So it might seem right
away that there is no way to make any good progress on this problem.

However, it turns out that it is possible to get results if
instead of talking about any polynomial, we only talk about
**homogeneous** polynomials. These are polynomials in several
variables
with the property that if you look at every term individually, then adding
up the degrees of the variables in the term always gives you the same
number. This number is called the **degree of the polynomial**. For
example, one homogeneous polynomial of degree 5 would be

I'm interested in finding values of the variables which are integers
and make the form equal to zero. If you think about it for a minute,
you'll see that this is easy. For example, in the polynomial above, we
can just set all of the variables equal to zero. Even better, if you
think about the definition, you'll see that this trick works for
__any__ homogeneous polynomial. This is because every term has to
contain
at least one variable. The solution in which all of the variables are set
to zero is called the **trivial zero** of a form (or system). I am
interested in determining whether there are **nontrivial zeros**, in
other words, zeros with at least one variable not equal to zero.

Now, there are certainly forms with no nontrivial zeros. For
example, the form

The amazing thing is that these are the only types of problems that
can arise. It turns out that if __the degree of a form is odd__
(because odd powers can be both positive and negative) and __there are
enough variables__, then the form always has a nontrivial zero. In
fact, a similar statement is true for systems of forms too. If the degree
of each form is odd (the forms can have different degrees) and there are
enough total variables, then the system has a nontrivial zero. In other
words, you can assign integers to all the variables with at least one
variable nonzero such that __all__ of the forms are equal to
zero.

Now, the interesting question to ask is "how many variables are
enough?" This is the question I try to answer. However, instead of
using regular integers like -3, 0, 5 and so on, I try to find zeros of
the forms in what are called **p-adic integers**. Before I tell you
what that means, let me just say first that it is possible to show that if
there is a nontrivial zero in ordinary integers, then there is also a
nontrivial zero in *p*-adic integers. Also, in order to use one of
the
standard methods to attack this problem in regular integers, it is often
necessary to first have an answer that works for *p*-adic
integers. So it's often important to know about *p*-adic zeros
before we can find out about zeros in ordinary integers.

So what is a *p*-adic integer? Well, they are pretty complicated to
define, so I won't try to give you the definition here. Fortunately however,
I can tell you what it means to say that a polynomial has a zero in
*p*-adic integers without defining these integers themselves. First,
let me say that the "*p*" in "*p*-adic" represents a prime
number, and there is a set of *p*-adic integers for every prime
*p*. So there is a set of 2-adic integers, a different set of 3-adic
integers, other sets of 5-adic integers, 7-adic integers, 11-adic integers and
so on. So when we have an equation to solve, we want to pick
the value of *p* first and then solve the equation in *p*-adic
integers for that specific value of *p*.

Now, suppose that *f(x)* is a polynomial and *p* is a prime
number (I'll do a couple of specific examples in a minute). Then to say
that *f(x)* has a zero in the *p*-adic integers means that you
can find values of *x* that make *f(x)* divisible by every power of
*p*. For example, suppose that *f(x) = x ^{2} + 2* and we
want to know whether

Now let's look at a different example. We'll still let *f(x) = x ^{2} +
2*, but now we'll see if there is a zero of this in the 5-adic integers. First,
we need to see if we can make

These two examples already show some interesting things about *p*-adic integers.
The first example shows that the *p*-adic integers are not the same as the
ordinary integers, since the equation *x ^{2} + 2 = 0* has no solutions
in the ordinary integers but does have solutions in the 3-adic integers. The second
example shows that the

It turns out that there are many differences between the *p*-adic integers
and the regular integers. For example, the *p*-adic
integers
are not ordered in the way that regular integers
are. In other words, given two *p*-adic integers, there is no
way in
which you can say that one of them is "greater" than the other. This
concept of order just doesn't make sense in the *p*-adic world. A
consequence of this is that in the *p*-adic world, there is no
concept of
"positive" and "negative" numbers. After all, you normally think of
positive
numbers as those that are greater than zero. If you can't define what
"greater than" means, then how can you define positive?

Another way in which *p*-adic integers are different is in
when forms
have nontrivial zeros. It turns out that a form of __any__ degree
has a nontrivial *p*-adic zero as long as there are enough
variables, and
the same is true for systems of forms. So in the *p*-adic world,
the degrees don't have to be odd anymore. The exact questions that I work
on are about how many variables are needed to ensure that (systems of)
forms have nontrivial *p*-adic zeros.

In particular, most of my work has dealt with **diagonal** forms.
That is, forms with no cross-terms. So a typical diagonal form might look
like

One of the results that I have proven is that if there are many forms
(say the number of forms is *R*) and all of the degrees
of the forms are the same (say they're all equal to *d*), then
there's a nontrivial *p*-adic zero, no matter what *p* is, as
long as there
are at least 4*R*^{2}*d*^{2} variables.
(Artin's conjecture says
that *Rd*^{2} + 1 variables should be enough, but I don't
know how to prove
that yet.) If *d* is even and large then
this is the
best-known bound. If *d* is odd, or if *d* is even and
small, then other people have found better bounds than mine. That
is, they
have found smaller numbers of variables that are guaranteed to work. I
won't say exactly what "large" means here, but if you know the value of
*R*, then you can figure out what "large" means for that value of
*R*. A particular value of *d* might be large for some
values of *R* but small for other values of *R*.

Another theorem that I have proven is that if there are
two (and only two) diagonal forms, the degrees are both odd, and one of
the degrees is at least 31, then Artin's conjecture is true. As
you might
guess from this result, it is easier to deal with odd degrees than even
degrees. One project that I am currently working on is to prove this
theorem
without the restriction that one of the degrees is at least 31.

I have proven several other theorems related to solving systems of forms
in *p*-adic integers, but they are a bit more complicated to explain and
my goal here is to attempt to keep this description reasonably straightforward. So
I won't mention these other theorems here. If you are interested, please check out
the Papers and Preprints section of my website. There you can
find abstracts for each of the research papers I have written, and can even look
at the papers themselves.

I hope that you have enjoyed reading this explanation of my research,
and that you have found it reasonably clear. As I said at the beginning,
if some parts are confusing, please feel free to email me about them.
If you are interested in learning more about this subject, I would be
happy to talk to you about it. Or, you could try reading a
more advanced account of this field. My
research
statement focuses on the specific problems that I work on.
Another good
source, which covers much more, is the article "Diophantine problems in
many variables: the role of
additive number theory" by Trevor Wooley, which appears in the book Topics
in Number Theory (Kluwer Academic Publishers, 1999).