<span class=strong>Reception and Poster Session</span>
Monday, April 16, 2007 - 4:00pm - 6:00pm
- Computing Zeta Functions of Sparse, Nondegenerate Hypersurfaces
John Voight (University of Minnesota, Twin Cities)
We discuss the efficient computation of the zeta function of a
hypersurface X defined over a finite field k. The zeta function
Z(X,T) of X is the exponential generating series for the number of
points on X defined over k and its finite extensions; Z(X,T) is a
rational function in T. Zeta functions arise naturally in coding
theory, arithmetic geometry, complexity theory, and many other areas.
We consider the situation when X is affine, toric, or projective, when
its defining equation f is nondegenerate (a condition which implies
that X is smooth and which holds outside of a codimension 1 subset),
and when f is sparse (consisting of few monomials). In this
situation, one can use Dwork cohomology to give a very efficient
method of computing Z(X,T). We exhibit this method and provide
- Decoding Linear Codes via Solving Systems of Polynomial Equations
Stanislav Bulygin (Universität Kaiserslautern)
This is a joint work of the presenter and Ruud Pellikaan (Technical University of Eindhoven, Netherlands). We
investigate a question of decoding linear codes, in particular up to half the minimum distance. We propose a method based on solving systems of polynomial equations over a finite field. Such a method was already quite successfully applied for cyclic codes earlier, we move on to the general case. We concentrate on solving the systems that emerge with the use of Groebner bases. The poster reflects a theoretical framework, the main result (the system we want to solve has a unique solution, etc.), and also some results on complexity estimation and experimental results.
- Nonextendibility of Mutually Unbiased Bases
Meera Sitharam (University of Florida)
Two orthonormal bases of a finite dimensional Hilbert space are
mutually unbiased if for every pair of vectors one from each basis,
the modulus of the standard inner product is the same.
The problem of determining bounds on the maximum number of
such bases (MUBs) for a fixed dimension is an important open problem
of 20 years that has been shown to arise in different guises in many
areas of mathematics.
The most common application of MUBs is found in quantum cryptography
where MUBs are the quantum states used in many protocols.
Here we consider 2 questions and provide preliminary results
- When is a given set of MUBs non-extendible? I.e.,
characterize mutually unbiased sets
M1,M2,..,Mk of for which there is no
basis M unbiased to M1,..,Mk?
- Characterize families of bases where non-extendibility
of a set of MUBs imply maximality, and are there natural
families of bases with this property. I.e, within such a family of
bases, a greedy method for constructing MUBs would guarantee
a maximal cardinality MUB collection.
This is joint work with Oscar Boykin and Mohamad Tarifi
at the University of Florida.
- When is a given set of MUBs non-extendible? I.e.,
- Metric Structure of Linear Codes
Diego Ruano (Universität Kaiserslautern)
We use the study of bilinear forms over a finite field to give a decomposition of the linear codes similar to the one in [D. Ruano: On the structure of generalized toric codes, arXiv:cs.IT/0611010, 2006] for generalized toric codes. Such decomposition, called geometric decomposition of a linear code and which can be obtained in a constructive way, allows to express easily the dual of a linear code and gives a new paradigm for linear codes. The proofs for characteristic 2 are different, but they will be developed parallel.
- A Complexity-reduced Interpolation Algorithm for Soft-decision Decoding of Reed-Solomon Codes
Kwankyu Lee (Korea Institute for Advanced Study (KIAS))
Soon after Lee and O'Sullivan proposed a new interpolation algorithm for
algebraic soft-decision decoding of Reed-Solomon codes, there have been some
attempts to apply a coordinate transformation technique to the new
algorithm, with a remarkable complexity reducing effect. We propose a
conceptually simple way of applying the transformation technique to the
- New Fewnomial Upper Bounds
Frank Sottile (Texas A & M University)
In 1980, Askold Khovanskii established his fewnomial bound
for the number of real solutions to a system of polynomials,
showing that the complexity of the set of real solutions to
a system of polynomials depends upon the number of monomials
and not on the degree. This fundamental finiteness result
in real algebraic geometry is believed to be unrealistically large.
I will Describe joint work with Frederic Bihan on a new
fewnomial bound which is substantially lower than Khovanskii's
bound and asymptotically optimal. This bound is obtained by
first reducing a given system to a Gale system, and then
bounding the number of solutions to a Gale system. Like
Khovanskii's bound, this bound is the product of an exponential
function and a polynomial in the dimension, with the exponents
in both terms depending upon the number of monomials. In our
bound, the exponents are smaller than in Khovanskii's.
- Descartes Rule for Complete Fields and Arbitrary Codimension
J. Maurice Rojas (Texas A & M University)
We discuss an extension of Descartes' Rule (for univariate
polynomials over the real numbers) to arbitrary k x n polynomial
systems over a field L which is either the real numbers or the
p-adic rationals. Over R, our extension fails for certain systems,
but we can characterize precise regions (in the coefficient space)
where our extension holds. Over the p-adic rationals, our extension
holds over even larger regions in the coefficient space. We then
conclude with a conjecture on what optimal fewnomial estimates should
like in general.
- List Decoding BCH Codes
Philip Busse (University of Kentucky)
List decoding Reed-Solomon codes via an interpolation and root-finding algorithm was pioneered by Madhu Sudan in the mid to late 1990's. In 2006, Kwankyu Lee and Michael O'Sullivan developed a Gröbner basis based implementation of the interpolation algorithm, which they showed was an efficient generalization of the Berlekamp-Massey Algorithm.
We propose to apply these ideas to list decoding BCH codes by
presenting BCH codes as subfield subcodes of generalized Reed-Solomon
codes and to explore the resulting algorithm. In particular, we seek ways to optimize the new algorithm and to compare it with the Berlekamp-Massey algorithm.
- The Word Problem for a Class of Groups with Infinite Presentation: Amputationally Universal Problem in the BSS Model
Klaus Meer (Syddansk Universitet (University of Southern Denmark))
Joint work with Martin Ziegler.
The word problem for discrete groups is well-known to be undecidable
by a Turing Machine; more precisely, it is reducible both to and from
and thus equivalent to the discrete Halting Problem.
The present work introduces and studies a real extension of
the word problem for a certain class of groups which are presented
as quotient groups of a free group and a normal subgroup.
The main result establishes the word problem for such groups to be
computationally equivalent to the Halting Problem for BSS machines
over the reals.
It thus provides the first non-trivial example of a concrete
problem that is computationally universal for the BSS model.
- NP, coNP and the Nullstellensatz: Lower Bounds for Stable Set and graph Coloring Nullstellensatze
Susan Margulies (University of California)
Systems of polynomial equations over the complex numbers can be used
to characterize NP-Complete graph-theoretic decision problems. From
the point of view of computer algebra and symbolic computation, these
are interesting polynomial systems because they are provably hard:
solving them is as hard as solving the underlying NP-Complete problem.
Furthermore, unless NP = coNP, there must exist infinite instances of
these infeasible systems whose Hilbert Nullstellensatz certificates
grow with respect to the underlying graphs.