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\centerline{\bf Gr\"obner Deformations of Hypergeometric Differential
Equations}

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\centerline{Bernd Sturmfels}
\centerline{Department of Mathematics, UC Berkeley}
\centerline{Berkeley, CA 94720, U.S.A }

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\noindent
This lecture gives an elementary introduction to a book with the above 
title, written  jointly with  Mutsumi Saito and Nobuki Takayama
and to appear shortly in the Springer Series
{\sl Algorithms and Computation in Mathematics}.
The current manuscript including a long list of references
can be found by surfing my home page \
{\sl http://math.berkeley.edu/\~{ }bernd/}.

One of our goals is to provide symbolic algorithms for 
constructing holomorphic solutions
$f(x_1,\ldots,x_n)$ to any system of linear partial differential
equations with polynomial coefficients. Such a differential system 
is represented by  a  left ideal $I$ in the Weyl algebra
$$ D \quad = \quad {\bf C} \langle x_1, \ldots, x_n,
\partial_1, \ldots, \partial_n \rangle . $$
By a {\it Gr\"obner deformation} of $I$ we mean an initial
ideal $\,in_{(-w,w)}(I) \subset D \,$ with respect to some
generic weight vector $w = (w_1,\ldots,w_n) \in {\bf R}^n$.
Here the variable $x_i$ has the weight $w_i$, and the
operator $\partial_i$ has the weight $-w_i$, so as to
respect the {\it product rule of calculus}:
$$ \partial_i \cdot x_i \quad = \quad x_i \cdot \partial_i + 1. $$
Using familiar techniques from computational commutative algebra,
one can determine an explicit solution basis for
the Gr\"obner deformation $\,in_{(-w,w)}(I)$. The big
problem is to extend it to a solution basis of $I$.
We solve this problem under the natural hypothesis that
the given $D$-ideal $I$ is {\it regular holonomic}.
Our main interest
lies in the systems of hypergeometric
differential equations introduced by Gel'fand, Kapranov and Zelevinski
in the 1980's. Here is a simple, but important, example of
a hypergeometric system for $n=3$:
$$ I \quad = \quad
D \cdot \bigl\{\,
\partial_1 \partial_3 - \partial_2^2, \,
x_1  \partial_1 + 
x_2  \partial_2 + 
x_3  \partial_3, \,
x_2  \partial_2 + 
2 x_3  \partial_3 - 1 \,\bigr\}. $$
If $w = (1,0,0)$ then the Gr\"obner deformation
of the hypergeometric system $I$ equals
$$ in_{(-w,w)}(I) \quad = \quad
D \cdot \bigl\{\,
\partial_1 \partial_3 \,, \,
x_1  \partial_1 + 
x_2  \partial_2 + 
x_3  \partial_3, \,
x_2  \partial_2 + 
2 x_3  \partial_3 - 1 \,\bigr\}.  $$
It is quite easy to see that the space of
solutions to $in_{(-w,w)}(I)$ is spanned
by $x_2/x_1$ and $x_3/x_2$. Starting from these
two Laurent monomials as $w$-lowest terms, our
algorithm constructs two linearly independent
Laurent series solutions to the original system $I$, namely,
$$ - { x_2 \over 2 x_1} \pm \biggl(
 { x_2 \over 2 x_1} \, - \sum_{m=0}^\infty
{1 \over m+1} { 2m \choose m} { x_1^m x_3^{m+1} \over
x_2^{2m+1} } \biggr)
\quad = \quad
{ -x_2 \pm \sqrt{ x_2^2 - 4 x_1 x_3} \over 2 x_1}
$$
This is the familiar {\it quadratic formula}
for expressing the two zeros of a quadratic polynomial
$\,p(z) \, = \, x_1 z^2 +  x_2 z +  x_3 \,$ in terms of its
three coefficients.  Can you write down the
hypergeometric differential equations  which encode the five roots
of the general quintic
$$ q(z) \quad = \quad 
x_1 z^5 + 
x_2 z^4 + 
x_3 z^3 + 
x_4 z^2 + 
x_5 z +
x_6 \quad ? $$

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