Computational Algebra Seminar: Fleischmann, Pohst

Speaker: Peter Fleischmann (Kent) 
Title: Some Aspects of Modular Invariant Theory of Finite Groups 
Time & Place: 2:30-3:30pm, Thursday 14 February, Carslaw 535.  

Abstract: Let $k$ be a field and $G$ a finite group, acting on the polynomial ring
$A:=k[x_1,\dots,x_d]$ by graded $k$-algebra automorphisms.  The ring of invariants
$A^G:=\{f\in A\ |\ g(f)=f\}$ is the main object of study in Invariant Theory.  The
theory is very well developed in the ``classical case", where the characteristic of $k$
is zero, but far less so in the case of positive characteristic, in particular the
``modular case", where the characteristic divides the group order $|G|$.  

In that case there are open questions about the constructive complexity of $A^G$,
measured by degree bounds for generators, and about the structural complexity, measured
by the depth (=length of maximal regular sequence, or ``cohomological co-dimension) of
$A^G$ as a module over a homogeneous system of parameters.  

In my talk I will, after a brief introduction, report on some recent results dealing
with both types of questions.  

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Speaker: Michael Pohst (Berlin) 
Title: On Solving Diophantine Equations over Global Fields 
Time & Place: 4:00-5:00pm, Thursday 14 February, Carslaw 535.  

Abstract: We present methods for the resolution of decomposable form equations over
global fields.  In general, those equations are reduced to unit equations.  Algorithms
for solving the latter differ substantially in the number and function field case.  Thue
and norm form equations will be discussed in greater detail.  Also, the fastest known
method for computing all integral points on Mordell curves y^2=x^3+k will be presented.