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A positive definite matrix is called logarithmically sparse if its matrix logarithm has many zero entries. Such matrices play a significant role in high-dimensional statistics and semidefinite optimization. In this paper, logarithmically sparse matrices are studied from the point of view of computational algebraic geometry: we present a formula for the dimension of the Zariski closure of a set of matrices with a given logarithmic sparsity pattern, give a degree bound for this variety and develop implicitization algorithms that allow to find its defining equations. We illustrate our approach with numerous examples. [ABSTRACT FROM AUTHOR]

The aim of this paper is to propose a strategy to implement the Minimal Model Program in modern computer algebra systems. [ABSTRACT FROM AUTHOR]

My paper (Suzuki 2003) produced some computer routines in Magma (Bosma et al. J Symb Comp 24:235–265, 1997) for the numerical invariants of Fano 3-folds, and used them in particular to determine the maximum value f = 19 of the Fano index. As a byproduct of the research, extensive data associated with all possible sets of singular points of Fano 3-folds with Fano indices greater than or equal to 2 was obtained. Collaborative research with Gavin Brown developed an improved version of the Magma program. The data discussed above was added to the Graded Ring Data Base (Brown et al. 2015) at the University of Kent. Subsequently, GRDB, now located to the University of Warwick, recently modified its interface to accommodate additional conditions, facilitating a more refined selection of Fano manifolds. In this context, we focus on the inequality known as the Bogomolov stability bound. We present a list of candidates for Fano 3-folds that do not satisfy these conditions and propose the conjecture that they do not exist.This result has been independently obtained in Liu and Liu (2023). [ABSTRACT FROM AUTHOR]

This article is a survey on the topic of polynomial amoebas. We review results of papers written on the topic with an emphasis on its computational aspects. Polynomial amoebas have numerous applications in various domains of mathematics and physics. Computation of the amoeba for a given polynomial and describing its properties is in general a problem of high complexity. We overview existing algorithms for computing and depicting amoebas and geometrical objects associated with them, such as contours and spines. We review the latest software packages for computing polynomial amoebas and compare their functionality and performance. [ABSTRACT FROM AUTHOR]

We present a series of algorithms for computing geometric and representation-theoretic invariants of Calogero–Moser spaces and rational Cherednik algebras associated with complex reflection groups. In particular, we are concerned with Calogero–Moser families (which correspond to the C × -fixed points of the Calogero–Moser space) and cellular characters (a proposed generalization by Rouquier and the first author of Lusztig’s constructible characters based on a Galois covering of the Calogero–Moser space). To compute the former, we devised an algorithm for determining generators of the center of the rational Cherednik algebra (this algorithm has several further applications), and to compute the latter we developed an algorithmic approach to the construction of cellular characters via Gaudin operators. We have implemented all our algorithms in the Cherednik Algebra Magma Package by the second author and used this to confirm open conjectures in several new cases. As an interesting application in birational geometry we are able to determine for many exceptional complex reflection groups the chamber decomposition of the movable cone of a Q -factorial terminalization (and thus the number of non-isomorphic relative minimal models) of the associated symplectic singularity. Making possible these computations was also a source of inspiration for the first author to propose several conjectures about the geometry of Calogero–Moser spaces (cohomology, fixed points, symplectic leaves), often in relation with the representation theory of finite reductive groups. [ABSTRACT FROM AUTHOR]

A projective variety X ⊂ P r is said to satisfy property QR(k) if its homogeneous ideal can be generated by quadratic polynomials of rank at most k. We define the rank index of X to be the smallest integer k such that X satisfies property QR(k). Many classical varieties, such as Segre-Veronese embeddings, rational normal scrolls and curves of high degree, satisfy property QR(4). Recently, it is shown in [19] that every Veronese embedding has rank index 3 if the base field has characteristic ≠ 2 , 3 . In this paper, we find the rank index of X when it is some other classical projective variety such as rational normal scrolls, Segre varieties, Plücker embeddings of the Grassmannians of lines and del Pezzo varieties. [ABSTRACT FROM AUTHOR]

Let W ⊂ P 13 be the image of the rational map defined by the linear system of the sextic surfaces of P 3 having double points along the edges of a tetrahedron. Let L be the linear system of the hyperplane sections of W. It is known that a general S ∈ L is an Enriques surface. The aim of this paper is to study the sublinear system L ∙ ⊂ L of the hyperplane sections of W having a triple point at a general point w ∈ W . We will show that a general element of L ∙ is birational to an elliptic ruled surface and that the image of W via the rational map defined by L ∙ is a cubic Del Pezzo surface Δ ⊂ P 3 with 4 nodes. Interestingly, this fact appears to be related to a conjecture of Castelnuovo. [ABSTRACT FROM AUTHOR]

Let (A , L , Θ n) be a dimension g abelian variety together with a level n theta structure over a field k of odd characteristic. We thus denote by (θ i Θ L) (Z / n Z) g ∈ Γ (A , L) the associated standard basis. For a positive integer ℓ relatively prime to n and the characteristic of k, we study change of level algorithms which allow one to compute level ℓ n theta functions (θ i Θ L ℓ (x)) i ∈ (Z / ℓ n Z) g from the knowledge of level n theta functions (θ i Θ L (x)) (Z / n Z) g or vice versa. The classical duplication formulas are an example of change of level algorithm to go from level n to level 2n. The main result of this paper states that there exists an algorithm to go from level n to level ℓ n in O (n g ℓ 2 g) operations in k. We derive an algorithm to compute an isogeny f : A → B from the knowledge of (A , L , Θ n) and K ⊂ A [ ℓ ] isotropic for the Weil pairing which computes f(x) for x ∈ A (k) in O ((n ℓ) g) operations in k. We remark that this isogeny computation algorithm is of quasi-linear complexity in the size of K. [ABSTRACT FROM AUTHOR]

For real and complex homogeneous cubic polynomials in n + 1 variables, we prove that the Chow variety of products of linear forms is generically complex identifiable for all ranks up to the generic rank minus two. By integrating fundamental results of Oeding (Adv Math 231:1308–1326, 2012), Casarotti and Mella (J Eur Math Soc, 2021) and Torrance and Vannieuwenhoven (Trans Am Math Soc 374:4815–4838, 2021), the proof is reduced to only those cases in up to 103 variables. These remaining cases are proved using the Hessian criterion for tangential weak defectivity from Chiantini et al. (SIAM J Matrix Anal Appl 35(4):1265–1287, 2014). We also establish that the smooth loci of the real and complex Chow varieties are immersed minimal submanifolds in their usual ambient spaces. [ABSTRACT FROM AUTHOR]

We construct some new deformation families of four-dimensional Fano manifolds of index one in some known classes of Gorenstein formats. These families have explicit descriptions in terms of equations, defining their image under the anticanonical embedding in some weighted projective space. They also have relatively smaller anticanonical degree than most other known families of smooth Fano 4-folds. [ABSTRACT FROM AUTHOR]

We describe discrete restricted Boltzmann machines: probabilistic graphicalmodels with bipartite interactions between visible and hidden discretevariables. Examples are binary restricted Boltzmann machines and discrete naiveBayes models. We detail the inference functions and distributed representationsarising in these models in terms of configurations of projected products ofsimplices and normal fans of products of simplices. We bound the number ofhidden variables, depending on the cardinalities of their state spaces, forwhich these models can approximate any probability distribution on theirvisible states to any given accuracy. In addition, we use algebraic methods andcoding theory to compute their dimension.

We derive relations between theoretical properties of restricted Boltzmannmachines (RBMs), popular machine learning models which form the building blocksof deep learning models, and several natural notions from discrete mathematicsand convex geometry. We give implications and equivalences relatingRBM-representable probability distributions, perfectly reconstructible inputs,Hamming modes, zonotopes and zonosets, point configurations in hyperplanearrangements, linear threshold codes, and multi-covering numbers of hypercubes.As a motivating application, we prove results on the relative representationalpower of mixtures of product distributions and products of mixtures of pairs ofproduct distributions (RBMs) that formally justify widely held intuitions aboutdistributed representations. In particular, we show that a mixture of productsrequiring an exponentially larger number of parameters is needed to representthe probability distributions which can be obtained as products of mixtures.

We show that positivity (≥0) on R + n and on R n of real symmetric polynomials of degree at most p in n ≥ 2 variables is solvable by algorithms running in polynomial time in the number n of variables. For real symmetric quartics, we find discriminants which lead to the efficient algorithms QE4+ and QE4 running in O (n) time. We describe the Maple implementation of both algorithms, which are then used not only for testing concrete inequalities (with given numerical coefficients and number of variables), but also for proving symbolic inequalities. [ABSTRACT FROM AUTHOR]

We consider the problem of identifying a mixture of Gaussian distributions with the same unknown covariance matrix by their sequence of moments up to certain order. Our approach rests on studying the moment varieties obtained by taking special secants to the Gaussian moment varieties, defined by their natural polynomial parametrization in terms of the model parameters. When the order of the moments is at most three, we prove an analogue of the Alexander–Hirschowitz theorem classifying all cases of homoscedastic Gaussian mixtures that produce defective moment varieties. As a consequence, identifiability is determined when the number of mixed distributions is smaller than the dimension of the space. In the two-component setting, we provide a closed form solution for parameter recovery based on moments up to order four, while in the one-dimensional case we interpret the rank estimation problem in terms of secant varieties of rational normal curves. [ABSTRACT FROM AUTHOR]

In this article, we investigate Muirhead's classical system of differential operators for the hypergeometric function 1 F 1 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. [ABSTRACT FROM AUTHOR]

In the field of numerical algebraic geometry, positive-dimensional solution sets of systems of polynomial equations are described by witness sets. In this paper, we define multiprojective witness sets which encode the multidegree information of an irreducible multiprojective variety. Our main results generalise the regeneration solving procedure, a trace test, and numerical irreducible decomposition to the multiprojective case. Examples are included to demonstrate this new approach. [ABSTRACT FROM AUTHOR]

Let A be the local ring, at a singular point P, of an algebraic variety V ⊂ A k r + 1 of multiplicity e = e (A) > 1 . If V is a curve in Orecchia (Can Math Bull 24:423–431, 1981) P was said to be an ordinary singularity when V has e (simple) tangents at P or equivalently when the projectivized tangent cone Proj (G (A)) of V at P is reduced (in which case consists of e points). In this paper, we show that the definition of ordinary singularity has a natural extension to higher dimensional varieties, in the case in which P is an isolated singularity and the normalization A ¯ of A is regular. In fact, we define P to be an ordinary singularity if the projectivized tangent cone Proj (G (A)) of V at P is reduced, i.e. is a variety in P k r . We prove that an ordinary singularity has multilinear projectivized tangent cone that is a union of e linear varieties L 1 , ... , L e . In the case in which L 1 , ... , L e are in generic position, we show that the affine tangent cone is also reduced and then multilinear. Finally, we show how to construct wide classes of parametric varieties with regular normalization at a singular isolated ordinary point. [ABSTRACT FROM AUTHOR]

We compute the dimensions of all the secant varieties to the tangential varieties of all Segre–Veronese surfaces. We exploit the typical approach of computing the Hilbert function of special 0-dimensional schemes on projective plane by using a new degeneration technique. [ABSTRACT FROM AUTHOR]

The signature of a parametric curve is a sequence of tensors whose entries are iterated integrals. This construction is central to the theory of rough paths in stochastic analysis. It is examined here through the lens of algebraic geometry. We introduce varieties of signature tensors for both deterministic paths and random paths. For the former, we focus on piecewise linear paths, on polynomial paths, and on varieties derived from free nilpotent Lie groups. For the latter, we focus on Brownian motion and its mixtures.

Let L 1 , … , L s be line bundles on a smooth complex variety X ⊂ P r and let D 1 , … , D s be divisors on X such that D i represents L i . We give a probabilistic algorithm for computing the degree of intersections of polar classes which are in turn used for computing the Euler characteristic of linear combinations of L 1 , … , L s . The input consists of generators for the homogeneous ideals I X , I D i ⊂ C [ x 0 , … , x r ] defining X and D i . [ABSTRACT FROM AUTHOR]

The distortion varieties of a given projective variety are parametrized by duplicating coordinates and multiplying them with monomials. We study their degrees and defining equations. Exact formulas are obtained for the case of one-parameter distortions. These are based on Chow polytopes and Gröbner bases. Multi-parameter distortions are studied using tropical geometry. The motivation for distortion varieties comes from multi-view geometry in computer vision. Our theory furnishes a new framework for formulating and solving minimal problems for camera models with image distortion. [ABSTRACT FROM AUTHOR]

The Galois/monodromy group of a family of geometric problems or equations is a subtle invariant that encodes the structure of the solutions. We give numerical methods to compute the Galois group and study it when it is not the full symmetric group. One algorithm computes generators, while the other studies its structure as a permutation group. We illustrate these algorithms with examples using a Macaulay2 package we are developing that relies upon Bertini to perform monodromy computations. [ABSTRACT FROM AUTHOR]

We present algorithms for basic computations with monoids in finitely generated abelian groups such as monoid membership testing and computing an element of the conductor ideal. Applying them to Mori dream spaces, we obtain algorithms to test whether a Weil divisor class of a given Mori dream space is base point free, to compute generators of the monoid of base point free Cartier divisor classes and to test whether a Q -factorial Mori dream space with known canonical class fulfills Fujita's base point free conjecture or not. [ABSTRACT FROM AUTHOR]