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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 S-polynomial reduction?

  • Abstract: Gröbner basis computation usually requires many reductions of S-polynomials. These reductions are computationally expensive. Bruno Buchberger discovered that sometimes we can skip the reduction of some S-polynomials. 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 n-th 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 Wen-shin 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.Ben-Or 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 $p-1$, to a $t$- sparse polynomial modulo $p^k$, of degree bounded by $(p-1)p^{k-1}$, 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 Ben-Or and Tiwari.

    Whil making some progress on the general question raised by Ben-Or 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 non-singular 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 n-tuples 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 "worst-case" examples in computational algebra, such as the Mayr-Meyer 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 zero-dimensional 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.
11 October 2004