43 research outputs found
Separating algebras and finite reflection groups
A separating algebra is, roughly speaking, a subalgebra of the ring of
invariants whose elements distinguish between any two orbits that can be
distinguished using invariants. In this paper, we introduce a geometric notion
of separating algebra. This allows us to prove that only groups generated by
reflections may have polynomial separating algebras, and only groups generated
by bireflections may have complete intersection separating algebras.Comment: 12 pages, corrected yet another typ
The separating variety for the basic representations of the additive group
For a group acting on an affine variety , the separating variety is
the closed subvariety of encoding which points of are separated
by invariants. We concentrate on the indecomposable rational linear
representations of dimension of the additive group of a field of
characteristic zero, and decompose the separating variety into the union of
irreducible components. We show that if is odd, divisible by four, or equal
to two, the closure of the graph of the action, which has dimension , is
the only component of the separating variety. In the remaining cases, there is
a second irreducible component of dimension . We conclude that in these
cases, there are no polynomial separating algebras.Comment: 14 page
Finite separating sets and quasi-affine quotients
Nagata's famous counterexample to Hilbert's fourteenth problem shows that the
ring of invariants of an algebraic group action on an affine algebraic variety
is not always finitely generated. In some sense, however, invariant rings are
not far from affine. Indeed, invariant rings are always quasi-affine, and there
always exist finite separating sets. In this paper, we give a new method for
finding a quasi-affine variety on which the ring of regular functions is equal
to a given invariant ring, and we give a criterion to recognize separating
algebras. The method and criterion are used on some known examples and in a new
construction.Comment: I corrected typos and fixed the proof of Lemma 5.3, 11 page
Mapping toric varieties into low dimensional spaces
A smooth d-dimensional projective variety X can always be embedded into 2d + 1-dimensional space. In contrast, a singular variety may require an arbitrary large ambient space. If we relax our requirement and ask only that the map is injective, then any d-dimensional projective variety can be mapped injectively to 2d + 1-dimensional projective space. A natural question then arises: what is the minimal m such that a projective variety can be mapped injectively to m-dimensional projective space? In this paper we investigate this question for normal toric varieties, with our most complete results being for Segre-Veronese varieties
Separating invariants and local cohomology
The study of separating invariants is a recent trend in invariant theory. For a finite group acting linearly on a vector space, a separating set is a set of invariants whose elements separate the orbits of G. In some ways, separating sets often exhibit better behavior than generating sets for the ring of invariants. We investigate the least possible cardinality of a separating set for a given G-action. Our main result is a lower bound that generalizes the classical result of Serre that if the ring of invariants is polynomial then the group action must be generated by pseudoreflections. We find these bounds to be sharp in a wide range of examples
A finite separating set for Daigle and Freudenburg's counterexample to Hilbert's Fourteenth Problem
This paper gives the first explicit example of a finite separating set in an invariant ring which is not finitely generated, namely, for Daigle and Freudenburg's 5-dimensional counterexample to Hilbert's Fourteenth Problem
The geometry of sloppiness
The use of mathematical models in the sciences often involves the estimation of unknown parameter values from data. Sloppiness provides information about the uncertainty of this task. In this paper, we develop a precise mathematical foundation for sloppiness as initially introduced and define rigorously key concepts, such as `model manifold', in relation to concepts of structural identifiability. We redefine sloppiness conceptually as a comparison between the premetric on parameter space induced by measurement noise and a reference metric. This opens up the possibility of alternative quantification of sloppiness, beyond the standard use of the Fisher Information Matrix, which assumes that parameter space is equipped with the usual Euclidean metric and the measurement error is infinitesimal. Applications include parametric statistical models, explicit time dependent models, and ordinary differential equation models
Sampling real algebraic varieties for topological data analysis
Topological data analysis (TDA) provides tools for computing geometric and topological information about spaces from a finite sample of points. We present an adaptive algorithm for finding provably dense samples of points on real algebraic varieties given a set of defining polynomials for use as input to TDA. The algorithm utilizes methods from numerical algebraic geometry to give formal guarantees about the density of the sampling, and also employs geometric heuristics to reduce the size of the sample. As TDA methods consume significant computational resources that scale poorly in the number of sample points, our sampling minimization makes applying TDA methods more feasible. We provide a software package that implements the algorithm, and showcase it through several examples