5 research outputs found
Conormal Spaces and Whitney Stratifications
We describe a new algorithm for computing Whitney stratifications of complex
projective varieties. The main ingredients are (a) an algebraic criterion, due
to L\^e and Teissier, which reformulates Whitney regularity in terms of
conormal spaces and maps, and (b) a new interpretation of this conormal
criterion via primary decomposition, which can be practically implemented on a
computer. We show that this algorithm improves upon the existing state of the
art by several orders of magnitude, even for relatively small input varieties.
En route, we introduce related algorithms for efficiently stratifying affine
varieties, flags on a given variety, and algebraic maps.Comment: There is an error in the published version of the article (Found
Comput Math, 2022) which has been fixed in this update. Section 3 is entirely
new, but the downstream results Sections 4-6 remain largely the same. We have
also updated the Runtimes and Complexity estimates in Section 7. The def. of
the integral closure of an ideal has also been correcte
On irreducible components of real exponential hypersurfaces
Fix any algebraic extension of the field of
rationals. In this article we study exponential sets .
Such sets are described by the vanishing of so called exponential polynomials,
i.e., polynomials with coefficients from , in variables, and in
exponential functions. The complements of all exponential sets in form a Noethrian topology on , which we will call Zariski
topology. Let be a
polynomial such that The main result of this paper
states that, under Schanuel's conjecture over the reals, an exponential set
of codimension 1, for which the real algebraic set is irreducible
over , either is irreducible (with respect to the Zariski topology)
or every of its irreducible components of codimension 1 is a rational
hyperplane through the origin. The family of all possible hyperplanes is
determined by monomials of . In the case of a single exponential (i.e., when
is independent of ) stronger statements are shown which
are independent of Schanuel's conjecture.Comment: Some minor changes. Final version, to appear in Arnold Mathematical
Journa
Analytic Combinatorics in Several Variables: Effective Asymptotics and Lattice Path Enumeration
The field of analytic combinatorics, which studies the asymptotic behaviour
of sequences through analytic properties of their generating functions, has led
to the development of deep and powerful tools with applications across
mathematics and the natural sciences. In addition to the now classical
univariate theory, recent work in the study of analytic combinatorics in
several variables (ACSV) has shown how to derive asymptotics for the
coefficients of certain D-finite functions represented by diagonals of
multivariate rational functions. We give a pedagogical introduction to the
methods of ACSV from a computer algebra viewpoint, developing rigorous
algorithms and giving the first complexity results in this area under
conditions which are broadly satisfied. Furthermore, we give several new
applications of ACSV to the enumeration of lattice walks restricted to certain
regions. In addition to proving several open conjectures on the asymptotics of
such walks, a detailed study of lattice walk models with weighted steps is
undertaken.Comment: PhD thesis, University of Waterloo and ENS Lyon - 259 page