3,262 research outputs found
Polynomial Invariants for Affine Programs
We exhibit an algorithm to compute the strongest polynomial (or algebraic)
invariants that hold at each location of a given affine program (i.e., a
program having only non-deterministic (as opposed to conditional) branching and
all of whose assignments are given by affine expressions). Our main tool is an
algebraic result of independent interest: given a finite set of rational square
matrices of the same dimension, we show how to compute the Zariski closure of
the semigroup that they generate
On algebraic cellular automata
We investigate some general properties of algebraic cellular automata, i.e.,
cellular automata over groups whose alphabets are affine algebraic sets and
which are locally defined by regular maps. When the ground field is assumed to
be uncountable and algebraically closed, we prove that such cellular automata
always have a closed image with respect to the prodiscrete topology on the
space of configurations and that they are reversible as soon as they are
bijective
Regulous vector bundles
Among recently introduced new notions in real algebraic geometry is that of
regulous functions. Such functions form a foundation for the development of
regulous geometry. Several interesting results on regulous varieties and
regulous sheaves are already available. In this paper, we define and
investigate regulous vector bundles. We establish algebraic and geometric
properties of such vector bundles, and identify them with stratified-algebraic
vector bundles. Furthermore, using new results on curve-rational functions, we
characterize regulous vector bundles among families of vector spaces
parametrized by an affine regulous variety. We also study relationships between
regulous and topological vector bundles
On existence of double coset varieties
Let be a complex affine algebraic group and be closed
subgroups. The homogeneous space can be equipped with structure of a
smooth quasiprojective variety. The situation is different for double coset
varieties \dcosets{F}{G}{H}. In this paper we give examples showing that the
variety \dcosets{F}{G}{H} does not necessarily exist. We also address the
question of existence of \dcosets{F}{G}{H} in the category of constructible
spaces and show that under sufficiently general assumptions \dcosets{F}{G}{H}
does exist as a constructible space.Comment: 7 pages; this version incorporates additions suggested by a referee
of Colloquium Mathematicu
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