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 21 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 21 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.