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Symbolic Computation Seminar
North Carolina State University
Symbolic Computation Seminar
North Carolina State University
The Symbolic Computation Seminar takes place in 335 Harrelson Hall at 4pm unless otherwise noted.
Fall 2004 Schedule
Wednesday, September 1, 5:00  6:00pm (NOTE SPECIAL TIME )
 Ruriko Yoshida, Duke University

Short Rational Functions for Toric Algebra and Applications
 Abstract: In 1993 Barvinok gave an algorithm that counts lattice points in convex rational polyhedra in polynomial time when the dimension of the polytope is fixed via short rational functions. The main theorem on this talk concerns the computation of the toric ideal I_A of an integral n x d matrix A. We encode the binomials belonging to the toric ideal I_A associated with A using short rational functions. If we fix d and n, this representation allows us to compute a universal Groebner basis and the reduced Groebner basis of the ideal I_A, with respect to any term order, in polynomial time. We also derive a polynomial time algorithm for normal form computations which replaces in this new encoding the usual reductions of the division algorithm. This is joint work with J. De Loera, D. Haws, R. Hemmecke, P. Huggins and B. Sturmfels.
Wednesday, September 8
 Jack Perry, NCSU

When can we skip Spolynomial reduction?
 Abstract: Gröbner basis computation
usually requires many reductions of Spolynomials. These reductions are
computationally expensive. Bruno Buchberger discovered that sometimes
we can skip the reduction of some Spolynomials. Specifically, he gave
two criteria on their leading terms. A natural question is: are there
more?
This talk will give an affirmative answer. Further, we will provide the necessary and sufficient conditions on the leading terms when there are at most three polynomials. Generalizing the result to an arbitrary number of polynomials remains an open problem.
Wednesday, September 15
 David
Saunders
and Information Sciences, University
of Delaware

Integer matrix determinants and formulas for pi
 Abstract: Gosper gave a formula for Pi
which Bellard has used to design an
algorithm for computing the nth digit of pi without computing the
earlier ones. Almkvist, Krattenthaler, and Petersson have developed
a scheme for generating a family of similar formulas which involves
computing determinants of certain integer matrices. I will present
a summary of their approach and use it as motivation to discuss the
state of the art of determinant computation. Can there be a fully
automatic determinant program which, for all integer matrices, is
(nearly) the best?
Wednesday, October 6
 George E. Collins

Coefficients of Factor Polynomials
 Abstract: The max norm of an integral
polynomial is the maximum of the
absolute values of its coefficients. Let d be the max norm
of an integral polynomial A(x) and let e be the minimum of
the max norms of its irreducible factors. How large can
e/d be? Can it be greater than 1, greater than 2, arbitrarily
large? We explore these questions experimentally and obtain
some answers.
Wednesday, October 13
 Teresa Krick, Universidad de Buenos Aires

Sparse univariate polynomial interpolation
 Abstract: The motivation of this work
comes from interesting discussions with Wenshin Lee on her
work on sparse multivaritate interpolatin. She pointed out to me the
paper " A deterministic algorithm for sparse multivariate polynomial
interpolation", 1988,
where M.BenOr and P. Tiwari raised the question wheter one can
efficiently reconstruct
a univariate $t$sparse polynomial from its values in any set of $2t$
(positive) points.
This is work in progress , joint with Martin Avendano and Ariel Pacetti
, from the University of Buenos Aires.
The talk consists of two parts.
1) $t$sparse polynomials represented by evaluation programs: This representation allows to evaluate derivatives as well. Using this, there is a very simple matrix which eigenvalues are exactly the exponents arising in the $t$sparse polynomial. The coefficients are then recovered in the usual way, by solving the corresponding Vandermonde linear system.
2) Integer $t$sparse polynomials: I' ll present a Hensel type lemma that allows to lift an exact $t$sparse polynomial modulo $p$, of degree bounded by $p1$, to a $t$ sparse polynomial modulo $p^k$, of degree bounded by $(p1)p^{k1}$, from its values in $2t$ points of size bounded by $2p$. This Hensel interpolation allows in some cases to reconstruct an integer sparse polynomial from its values in more general and smaller height families of points than the ones used by BenOr and Tiwari.
Whil making some progress on the general question raised by BenOr and Tiwari, these results still do not give a definitive answer.
Wednesday, Ocotber 20, 4.30
pm, HA 335 (NOTE
SPECIAL TIME)
 Carlos D'Andrea,
University of California, Berkeley

The jacobian scheme of the discriminant
 Abstract: The discriminant of a binary
form of degree d is a polynomial
which appears naturally in algebraic geometry and invariant theory.
Geometrically, it encodes the variety of univariate polynomials with
multiple roots. The jacobian ideal of the discriminant contains more
precise information about the nature of these multiple roots.
In this talk, I will give a brief introduction to this topic and show the algebraic structure of the jacobian ideal.
This is a joint work with Jaydeep Chipalkatti.
Thursday, October 21, Computer Science Colloquium, 4:00pm, Withers 402A (NOTE SPECIAL TIME AND PLACE)
 Erich Kaltofen, NCSU

The Art of Symbolic Computation
 Abstract: Computation in the past 40
years has brought
us remarkable theory: Groebner bases, the Risch
integration algorithm, polynomial factorization
lattice basis reduction, hypergeometric summation algorithms,
sparse polynomial interpolation, etc. And it produced
remarkable software like Mathematica and Maple, which supplies
implementations of these algorithms to the masses.
Several system designers even became rich by selling their
software. As it turned out, a killer application
of computer algebra is high energy physics, where
a special purpose computer algebra system, SCHOONSHIP,
helped in work worthy of a Nobel Prize in physics
in 1999.
In my talk I will give glimpses what the descipline of Symbolic Computation is and what problems it tackles. In particular, I will discuss the use of heuristics (numerical, randomized, and algorithmic) that seem necessary to solve some of today's problems in geometric modeling and equation solving. Thus, we seem to have come full cycle (the discipline may have started in the 1960s at the MIT AI Lab), but with a twist that I shall explain.
Wednesday, October 27
 Zhendong Wan,
and Information Sciences, University
of Delaware

Exactly Solve Integer Linear Systems Using Numerical Methods
 Abstract: We present a new fast way to exactly solve nonsingular linear systems with integer coefficients using numerical methods. This is important both in practice and in theory. Our method is to find an initial approximate solution by using a fast numerical method, then approximate the solution amplified by a scalar, with integers. Update the amplified residual, and repeat these steps to refine the solution until high enough accuracy is achieved, and finally reconstruct the rational solution. We will expose the theoretical cost and show some experimental results.
Friday, October 29, 4:00pm,
Harrelson 330 (NOTE
SPECIAL TIME AND PLACE)
 Alicia Dickenstein,
Universidad de Buenos Aires

Binomial Complete Intersections (joint work with Eduardo Cattani)
 Abstract: A binomial ideal in the
polynomial ring R:=k[x_1,...,x_n], is an
ideal generated by binomials: a x^A  b x^B, where
A,B are nonnegative integer ntuples and a,b in k are non zero.
Binomial ideals are quite ubiquitous in very different contexts
particularly those involving toric geometry and its applications,
and in the study of semigroup algebras.
While binomial ideals are quite amenable to Groebner and standard
bases techniques, they also provide some of the "worstcase"
examples in computational algebra, such as the MayrMeyer ideals.
Thus, we are interested in algorithms that allow us to obtain
information about binomial ideals purely in terms of the
data defining them.
In this talk we will discuss how to determine when a binomial ideal is a zerodimensional complete intersection and, if so, how we may compute the total number of solutions and the total multiplicity of solutions in the coordinate axes. These problems arise naturally in the study of sparse discriminants and in the study of hypergeometric systems of differential equations.