1,721,041 research outputs found
Implicitisation and Parameterisation in Polynomial Functors
Full text linkIn earlier work, the second author showed that a closed subset of a
polynomial functor can always be defined by finitely many polynomial equations.
In follow-up work on -varieties,
Bik-Draisma-Eggermont-Snowden showed, among other things, that in
characteristic zero every such closed subset is the image of a morphism whose
domain is the product of a finite-dimensional affine variety and a polynomial
functor. In this paper, we show that both results can be made algorithmic:
there exists an algorithm that takes as input a morphism
into a polynomial functor and outputs finitely many equations defining the
closure of the image; and an algorithm that takes as
input a finite set of equations defining a closed subset of a polynomial
functor and outputs a morphism whose image is that closed subset.Comment: 22 page
Orthogonal and unitary tensor decomposition from an algebraic perspective
Full text linkWhile every matrix admits a singular value decomposition, in which the terms are pairwise orthogonal in a strong sense, higher-order tensors typically do not admit such an orthogonal decomposition. Those that do have attracted attention from theoretical computer science and scientific computing. We complement this existing body of literature with an algebro-geometric analysis of the set of orthogonally decomposable tensors. More specifically, we prove that they form a real-algebraic variety defined by polynomials of degree at most four. The exact degrees, and the corresponding polynomials, are different in each of three times two scenarios: ordinary, symmetric, or alternating tensors; and real-orthogonal versus complex-unitary. A key feature of our approach is a surprising connection between orthogonally decomposable tensors and semisimple algebras—associative in the ordinary and symmetric settings and of compact Lie type in the alternating setting
Tegenvoorbeelden voor het Helton-Nievermoeden
No full textHet Helton–Nievermoeden zegt dat elke convexe, semialgebraïsche verzameling semidefiniet
representeerbaar is. In december 2016 is dit vermoeden op spectaculaire wijze met
tegenvoorbeelden weerlegd door de Duitse wiskundige Claus Scheiderer. In dit artikel
beschrijft Jan Draisma de achtergronden van het vermoeden en de tegenvoorbeelden van
Scheiderer
Uniform determinantal representations
Full text linkThe problem of expressing a specific polynomial as the determinant of a
square matrix of affine-linear forms arises from algebraic geometry,
optimisation, complexity theory, and scientific computing. Motivated by recent
developments in this last area, we introduce the notion of a uniform
determinantal representation, not of a single polynomial but rather of all
polynomials in a given number of variables and of a given maximal degree. We
derive a lower bound on the size of the matrix, and present a construction
achieving that lower bound up to a constant factor as the number of variables
is fixed and the degree grows. This construction marks an improvement upon a
recent construction due to Plestenjak-Hochstenbach, and we investigate the
performance of new representations in their root-finding technique for
bivariate systems. Furthermore, we relate uniform determinantal representations
to vector spaces of singular matrices, and we conclude with a number of future
research directions.Comment: 23 pages, 3 figures, 4 table
Tropical ideals do not realise all Bergman fans
No full textTropical ideals are combinatorial objects that abstract the possible collections of subsets arising as the supports of all polynomials in an ideal. Every tropical
ideal has an associated tropical variety: a finite polyhedral complex equipped with
positive integral weights on its maximal cells. This leads to the realisability question,
ubiquitous in tropical geometry, of which weighted polyhedral complexes arise in this
manner. Using work of Las Vergnas on the non-existence of tensor products of matroids, we prove that there is no tropical ideal whose variety is the Bergman fan of
the direct sum of the Vámos matroid and the uniform matroid of rank two on three
elements, and in which all maximal cones have weight one
Partial correlation hypersurfaces in Gaussian graphical models
Full text linkWe derive a combinatorial sufficient condition for a partial correlation hypersurface in the parameter space of a directed Gaussian graphical model to be nonsingular, and speculate on whether this condition can be used in algorithms for learning the graph. Since the condition is fulfilled in the case of a complete DAG on any number of vertices, the result implies an affirmative answer to a question raised by Lin-Uhler-Sturmfels-Bühlmann
The irreducible control property in matrix groups
Full text linkThis paper concerns matrix decompositions in which the factors are restricted to lie in a closed subvariety of a matrix group. Such decompositions are of relevance in control theory: given a target matrix in the group, can it be decomposed as a product of elements in the subvarieties, in a given order? And if so, what can be said about the solution set to this problem? Can an irreducible curve of target matrices be lifted to an irreducible curve of factorisations? We show that under certain conditions, for a sufficiently long and complicated such sequence, the solution set is always irreducible, and we show that every connected matrix group has a sequence of one-parameter subgroups that satisfies these conditions, where the sequence has length less than 1.5 times the dimension of the group
Topological Noetherianity of polynomial functors II: base rings with Noetherian spectrum
Full text linkIn a previous paper, the third author proved that finite-degree polynomial functors over infinite fields are topologically Noetherian. In this paper, we prove that the same holds for polynomial functors from free R-modules to finitely generated R-modules, for any commutative ring R whose spectrum is Noetherian. As Erman–Sam–Snowden pointed out, when applying this with R=Z to direct sums of symmetric powers, one of their proofs of a conjecture by Stillman becomes characteristic-independent. Our paper advertises and further develops the beautiful but not so well-known machinery of polynomial laws. In particular, to any finitely generated R-module M we associate a topological space, which we show is Noetherian when Spec(R)
is; this is the degree-zero case of our result on polynomial functors
Catalan-many tropical morphisms to trees; Part I: Constructions
No full textWe investigate the tree gonality of a genus-g metric graph, defined as the minimum degree of a tropical morphism from any tropical modification of the metric graph to a metric tree. We give a combinatorial constructive proof that this number is at most ⌈g/2⌉+1, a fact whose proofs so far required an algebro-geometric detour via special divisors on curves. For even genus, the tropical morphism which realizes the bound belongs to a family of tropical morphisms that is pure of dimension 3g−3 and that has a generically finite-to-one map onto the moduli space of genus-g metric graphs. Our methods focus on the study of such families. This is part I in a series of two papers: in part I we fix the combinatorial type of the metric graph to show a bound on tree-gonality, while in part II we vary the combinatorial type and show that the number of tropical morphisms, counted with suitable multiplicities, is the same Catalan number that counts morphisms from a general genus-g curve to the projective line
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