Department of Mathematics Colloquium


Abstract 

Given a finite set of points in a projective
or Euclidean space, the "Exact Fitting Problem" determines the maximum
number of these points contained in some hyperplane. Besides the
applications in Optimization Theory or Computer Graphic Design and
Pattern Recognition, this problem is strongly related to important
questions in Coding Theory. If one considers any generating matrix of a
linear code and thinks of the columns of this matrix as the homogeneous
coordinates of points in a projective space (assuming that there are no
proportional nor zero columns), solving the Exact Fitting Problem for
these points also determines the minimum distance of the code. I will
present a Commutative Algebraic approach to solve this problem and I
will also show how some information contained in the graded minimal
free resolution of the ideal of the set of points constructed from the
columns of the generating matrix, gives lower bounds for the minimum
distance of the code. The talk will address to a general audience, with
a minimal background in Linear Algebra, and some knowledge in Abstract
Algebra (e.g., rings, modules, ideals, the ring of polynomials).
