917 research outputs found
Vector field construction of Segre sets
A CR generic real analytic CR manifold M carries two families of Segre
varieties and conjugate Segre varieties. We observe in this article that their
complexifications give rise to two families of foliations of the
complexification of M which coincide with the flow foliations induced by the
complexified CR (1,0) and (0,1) vector fields tangent to M. As an application,
we derive a new proof of the characterization of finite type in terms of Segre
sets.Comment: Completely rewritten version, 42 pages Sujclass: Complex Variable
Algebraic relations between solutions of Painlev\'e equations
We calculate model theoretic ranks of Painlev\'e equations in this article,
showing in particular, that any equation in any of the Painlev\'e families has
Morley rank one, extending results of Nagloo and Pillay (2011). We show that
the type of the generic solution of any equation in the second Painlev\'e
family is geometrically trivial, extending a result of Nagloo (2015).
We also establish the orthogonality of various pairs of equations in the
Painlev\'e families, showing at least generically, that all instances of
nonorthogonality between equations in the same Painlev\'e family come from
classically studied B{\"a}cklund transformations. For instance, we show that if
at least one of is transcendental, then is
nonorthogonal to if and only if or . Our results have concrete interpretations
in terms of characterizing the algebraic relations between solutions of
Painlev\'e equations. We give similar results for orthogonality relations
between equations in different Painlev\'e families, and formulate some general
questions which extend conjectures of Nagloo and Pillay (2011) on transcendence
and algebraic independence of solutions to Painlev\'e equations. We also apply
our analysis of ranks to establish some orthogonality results for pairs of
Painlev\'e equations from different families. For instance, we answer several
open questions of Nagloo (2016), and in the process answer a question of Boalch
(2012).Comment: This manuscript replaces and greatly expands a portion of
arXiv:1608.0475
Nonisomorphic Ordered Sets with Arbitrarily Many Ranks That Produce Equal Decks
We prove that for any there is a pair of
nonisomorphic ordered sets such that and have equal maximal
and minimal decks, equal neighborhood decks, and there are ranks such that for each the decks obtained by removing the points
of rank are equal. The ranks do not contain
extremal elements and at each of the other ranks there are elements whose
removal will produce isomorphic cards. Moreover, we show that such sets can be
constructed such that only for ranks and , both without extremal
elements, the decks obtained by removing the points of rank are not
equal.Comment: 30 pages, 6 figures, straight LaTe
Topological noetherianity for cubic polynomials
Let be the space of complex cubic polynomials in
infinitely many variables. We show that this space is
-noetherian, meaning that any
-stable Zariski closed subset is cut out by finitely many
orbits of equations. Our method relies on a careful analysis of an invariant of
cubics introduced here called q-rank. This result is motivated by recent work
in representation stability, especially the theory of twisted commutative
algebras. It is also connected to certain stability problems in commutative
algebra, such as Stillman's conjecture.Comment: 13 page
Uniqueness and minimal obstructions for tree-depth
A k-ranking of a graph G is a labeling of the vertices of G with values from
{1,...,k} such that any path joining two vertices with the same label contains
a vertex having a higher label. The tree-depth of G is the smallest value of k
for which a k-ranking of G exists. The graph G is k-critical if it has
tree-depth k and every proper minor of G has smaller tree-depth.
We establish partial results in support of two conjectures about the order
and maximum degree of k-critical graphs. As part of these results, we define a
graph G to be 1-unique if for every vertex v in G, there exists an optimal
ranking of G in which v is the unique vertex with label 1. We show that several
classes of k-critical graphs are 1-unique, and we conjecture that the property
holds for all k-critical graphs. Generalizing a previously known construction
for trees, we exhibit an inductive construction that uses 1-unique k-critical
graphs to generate large classes of critical graphs having a given tree-depth.Comment: 14 pages, 4 figure
Effective criteria for specific identifiability of tensors and forms
In applications where the tensor rank decomposition arises, one often relies
on its identifiability properties for interpreting the individual rank-
terms appearing in the decomposition. Several criteria for identifiability have
been proposed in the literature, however few results exist on how frequently
they are satisfied. We propose to call a criterion effective if it is satisfied
on a dense, open subset of the smallest semi-algebraic set enclosing the set of
rank- tensors. We analyze the effectiveness of Kruskal's criterion when it
is combined with reshaping. It is proved that this criterion is effective for
both real and complex tensors in its entire range of applicability, which is
usually much smaller than the smallest typical rank. Our proof explains when
reshaping-based algorithms for computing tensor rank decompositions may be
expected to recover the decomposition. Specializing the analysis to symmetric
tensors or forms reveals that the reshaped Kruskal criterion may even be
effective up to the smallest typical rank for some third, fourth and sixth
order symmetric tensors of small dimension as well as for binary forms of
degree at least three. We extended this result to symmetric tensors by analyzing the Hilbert function, resulting in a
criterion for symmetric identifiability that is effective up to symmetric rank
, which is optimal.Comment: 31 pages, 2 Macaulay2 code
- …