57 research outputs found
Coordinate-wise Powers of Algebraic Varieties
We introduce and study coordinate-wise powers of subvarieties of
, i.e. varieties arising from raising all points in a given
subvariety of to the -th power, coordinate by coordinate.
This corresponds to studying the image of a subvariety of under
the quotient of by the action of the finite group
. We determine the degree of coordinate-wise powers and
study their defining equations, particularly for hypersurfaces and linear
spaces. Applying these results, we compute the degree of the variety of
orthostochastic matrices and determine iterated dual and reciprocal varieties
of power sum hypersurfaces. We also establish a link between coordinate-wise
squares of linear spaces and the study of real symmetric matrices with a
degenerate eigenspectrum.Comment: 26 page
Towards classifying toric degenerations of cubic surfaces
We investigate the class of degenerations of smooth cubic surfaces which are
obtained from degenerating their Cox rings to toric algebras. More precisely,
we work in the spirit of Sturmfels and Xu who use the theory of Khovanskii
bases to determine toric degenerations of Del Pezzo surfaces of degree 4 and
who leave the question of classifying these degenerations in the degree 3 case
as an open problem. In order to carry out this classification we describe an
approach which is closely related to tropical geometry and present partial
results in this direction.Comment: v2: 21 pages, section 1 rewritten, added sections 6 and
Algebraic Analysis of the Hypergeometric Function 1F1 of a Matrix Argument
In this article, we investigate Muirhead's classical system of differential
operators for the hypergeometric function 1F1 of a matrix argument. We
formulate a conjecture for the combinatorial structure of the characteristic
variety of its Weyl closure which is both supported by computational evidence
as well as theoretical considerations. In particular, we determine the singular
locus of this system.Comment: 26 pages. v2 changes: included Appendix discussing the singular locus
for degenerate parameter
Towards classifying toric degenerations of cubic surfaces
We investigate the class of degenerations of smooth cubic surfaces which are obtained from degenerating their Cox rings to toric algebras. More precisely, we work in the spirit of Sturmfels and Xu who use the theory of Khovanskii bases to determine toric degenerations of Del Pezzo surfaces of degree 4 and who leave the question of classifying these degenerations in the degree 3 case as an open problem. In order to carry out this classification we describe an approach which is closely related to tropical geometry and present partial results in this direction.
 
Tautological systems, homogeneous spaces and the holonomic rank problem
Many GKZ-systems that arise from a geometric setting can be endowed with the
structure of mixed Hodge modules. We generalize this fundamental result to the
tautological systems associated to homogeneous spaces by giving a functorial
construction for them. As an application, we solve the holonomic rank problem
for such tautological systems in full generality
Rational invariants of even ternary forms under the orthogonal group
In this article we determine a generating set of rational invariants of
minimal cardinality for the action of the orthogonal group on
the space of ternary forms of even degree . The
construction relies on two key ingredients: On one hand, the Slice Lemma allows
us to reduce the problem to dermining the invariants for the action on a
subspace of the finite subgroup of signed permutations. On the
other hand, our construction relies in a fundamental way on specific bases of
harmonic polynomials. These bases provide maps with prescribed
-equivariance properties. Our explicit construction of these
bases should be relevant well beyond the scope of this paper. The expression of
the -invariants can then be given in a compact form as the
composition of two equivariant maps. Instead of providing (cumbersome) explicit
expressions for the -invariants, we provide efficient algorithms
for their evaluation and rewriting. We also use the constructed
-invariants to determine the -orbit locus and
provide an algorithm for the inverse problem of finding an element in
with prescribed values for its invariants. These are
the computational issues relevant in brain imaging.Comment: v3 Changes: Reworked presentation of Neuroimaging application,
refinement of Definition 3.1. To appear in "Foundations of Computational
Mathematics
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