We prove a $p$-adic version of the Integral Geometry Formula for averaging the intersection of two $p$-adic projective algebraic sets. We apply this result to give bounds on the number of points in the modulo $p^m$ reduction of a projective set (reproving a result by Oesterl'e) and to the study of random $p$-adic polynomial systems of equations.
{"title":"p-Adic Integral Geometry","authors":"Avinash Kulkarni, A. Lerário","doi":"10.1137/19m1284737","DOIUrl":"https://doi.org/10.1137/19m1284737","url":null,"abstract":"We prove a $p$-adic version of the Integral Geometry Formula for averaging the intersection of two $p$-adic projective algebraic sets. We apply this result to give bounds on the number of points in the modulo $p^m$ reduction of a projective set (reproving a result by Oesterl'e) and to the study of random $p$-adic polynomial systems of equations.","PeriodicalId":48489,"journal":{"name":"SIAM Journal on Applied Algebra and Geometry","volume":"1 1","pages":"28-59"},"PeriodicalIF":1.2,"publicationDate":"2019-08-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83706808","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study the structure of the set of algebraic curvature operators satisfying a sectional curvature bound under the light of the emerging field of Convex Algebraic Geometry. More precisely, we determine in which dimensions $n$ this convex semialgebraic set is a spectrahedron or a spectrahedral shadow; in particular, for $ngeq5$, these give new counter-examples to the Helton--Nie Conjecture. Moreover, efficient algorithms are provided if $n=4$ to test membership in such a set. For $ngeq5$, algorithms using semidefinite programming are obtained from hierarchies of inner approximations by spectrahedral shadows and outer relaxations by spectrahedra.
{"title":"Convex Algebraic Geometry of Curvature Operators","authors":"R. G. Bettiol, Mario Kummer, R. Mendes","doi":"10.1137/20M1350777","DOIUrl":"https://doi.org/10.1137/20M1350777","url":null,"abstract":"We study the structure of the set of algebraic curvature operators satisfying a sectional curvature bound under the light of the emerging field of Convex Algebraic Geometry. More precisely, we determine in which dimensions $n$ this convex semialgebraic set is a spectrahedron or a spectrahedral shadow; in particular, for $ngeq5$, these give new counter-examples to the Helton--Nie Conjecture. Moreover, efficient algorithms are provided if $n=4$ to test membership in such a set. For $ngeq5$, algorithms using semidefinite programming are obtained from hierarchies of inner approximations by spectrahedral shadows and outer relaxations by spectrahedra.","PeriodicalId":48489,"journal":{"name":"SIAM Journal on Applied Algebra and Geometry","volume":"10 1","pages":"200-228"},"PeriodicalIF":1.2,"publicationDate":"2019-08-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72970617","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Kippenhahn's Theorem asserts that the numerical range of a matrix is the convex hull of a certain algebraic curve. Here, we show that the joint numerical range of finitely many hermitian matrices is similarly the convex hull of a semi-algebraic set. We discuss an analogous statement regarding the dual convex cone to a hyperbolicity cone and prove that the class of convex bases of these dual cones is closed under linear operations. The result offers a new geometric method to analyze quantum states.
{"title":"Kippenhahn's Theorem for Joint Numerical Ranges and Quantum States","authors":"D. Plaumann, Rainer Sinn, S. Weis","doi":"10.1137/19M1286578","DOIUrl":"https://doi.org/10.1137/19M1286578","url":null,"abstract":"Kippenhahn's Theorem asserts that the numerical range of a matrix is the convex hull of a certain algebraic curve. Here, we show that the joint numerical range of finitely many hermitian matrices is similarly the convex hull of a semi-algebraic set. We discuss an analogous statement regarding the dual convex cone to a hyperbolicity cone and prove that the class of convex bases of these dual cones is closed under linear operations. The result offers a new geometric method to analyze quantum states.","PeriodicalId":48489,"journal":{"name":"SIAM Journal on Applied Algebra and Geometry","volume":"110 1","pages":"86-113"},"PeriodicalIF":1.2,"publicationDate":"2019-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74199474","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this article, we explore a connection between nonnegativity, the theory of A-discriminants, and tropical geometry. We show that the algebraic strata of the boundary of the sonc cone are parametrized by families of tropical hypersurfaces. Each strata is contained in a rational variety called a positive discriminant. As an application, we characterization generic support sets for which the sonc cone is equal to the sparse nonnegativity cone, and we give a complete description of the semi-algebraic stratification of the boundary of the sonc cone in the univariate case.
{"title":"The Algebraic Boundary of the Sonc-Cone","authors":"Jens Forsgård, T. Wolff","doi":"10.1137/20m1325484","DOIUrl":"https://doi.org/10.1137/20m1325484","url":null,"abstract":"In this article, we explore a connection between nonnegativity, the theory of A-discriminants, and tropical geometry. We show that the algebraic strata of the boundary of the sonc cone are parametrized by families of tropical hypersurfaces. Each strata is contained in a rational variety called a positive discriminant. As an application, we characterization generic support sets for which the sonc cone is equal to the sparse nonnegativity cone, and we give a complete description of the semi-algebraic stratification of the boundary of the sonc cone in the univariate case.","PeriodicalId":48489,"journal":{"name":"SIAM Journal on Applied Algebra and Geometry","volume":"1 1","pages":"468-502"},"PeriodicalIF":1.2,"publicationDate":"2019-05-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75364295","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The matroid-based valuation conjecture of Ostrovsky and Paes Leme states that all gross substitutes valuations on $n$ items can be produced from merging and endowments of weighted ranks of matroids defined on at most $m(n)$ items. We show that if $m(n) = n$, then this statement holds for $n leq 3$ and fails for all $n geq 4$. In particular, the set of gross substitutes valuations on $n geq 4$ items is strictly larger than the set of matroid based valuations defined on the ground set $[n]$. Our proof uses tropical geometry, matroid theory and discrete convex analysis to explicitly construct a large family of counter-examples. It indicates that merging and endowment by themselves are poor operations to generate gross substitutes valuations. We also connect the general MBV conjecture and related questions to long-standing open problems in matroid theory, and conclude with open questions at the intersection of this field and economics.
{"title":"The Finite Matroid-Based Valuation Conjecture is False","authors":"N. Tran","doi":"10.1137/19M1304295","DOIUrl":"https://doi.org/10.1137/19M1304295","url":null,"abstract":"The matroid-based valuation conjecture of Ostrovsky and Paes Leme states that all gross substitutes valuations on $n$ items can be produced from merging and endowments of weighted ranks of matroids defined on at most $m(n)$ items. We show that if $m(n) = n$, then this statement holds for $n leq 3$ and fails for all $n geq 4$. In particular, the set of gross substitutes valuations on $n geq 4$ items is strictly larger than the set of matroid based valuations defined on the ground set $[n]$. Our proof uses tropical geometry, matroid theory and discrete convex analysis to explicitly construct a large family of counter-examples. It indicates that merging and endowment by themselves are poor operations to generate gross substitutes valuations. We also connect the general MBV conjecture and related questions to long-standing open problems in matroid theory, and conclude with open questions at the intersection of this field and economics.","PeriodicalId":48489,"journal":{"name":"SIAM Journal on Applied Algebra and Geometry","volume":"29 1","pages":"506-525"},"PeriodicalIF":1.2,"publicationDate":"2019-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78113700","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper we characterize the polynomials $f$ over a finite field $F$ satisfying the following property: there exists an extension field $L$ of $F$ such that for any positive integer $ell$ less than or equal to the degree of $f$, there exists $t_0$ in $L$ with the property that the polynomial $f-t_0$ has an irreducible factor in $L[x]$ of degree $ell$. This result is then used to progress to the last step which is needed to remove the heuristic from one of the quasi-polynomial time algorithms for discrete logarithm problems (DLPs) in small characteristic. Our method is general and can be used to tackle similar problems which involve factorization patterns of polynomials over finite fields.
{"title":"On the Selection of Polynomials for the DLP Quasi-Polynomial Time Algorithm for Finite Fields of Small Characteristic","authors":"Giacomo Micheli","doi":"10.1137/18M1177196","DOIUrl":"https://doi.org/10.1137/18M1177196","url":null,"abstract":"In this paper we characterize the polynomials $f$ over a finite field $F$ satisfying the following property: there exists an extension field $L$ of $F$ such that for any positive integer $ell$ less than or equal to the degree of $f$, there exists $t_0$ in $L$ with the property that the polynomial $f-t_0$ has an irreducible factor in $L[x]$ of degree $ell$. This result is then used to progress to the last step which is needed to remove the heuristic from one of the quasi-polynomial time algorithms for discrete logarithm problems (DLPs) in small characteristic. Our method is general and can be used to tackle similar problems which involve factorization patterns of polynomials over finite fields.","PeriodicalId":48489,"journal":{"name":"SIAM Journal on Applied Algebra and Geometry","volume":"4 1","pages":"256-265"},"PeriodicalIF":1.2,"publicationDate":"2019-04-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84641706","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A bottleneck of a smooth algebraic variety $X subset mathbb{C}^n$ is a pair of distinct points $(x,y) in X$ such that the Euclidean normal spaces at $x$ and $y$ contain the line spanned by $x$ and $y$. The narrowness of bottlenecks is a fundamental complexity measure in the algebraic geometry of data. In this paper we study the number of bottlenecks of affine and projective varieties, which we call the bottleneck degree. The bottleneck degree is a measure of the complexity of computing all bottlenecks of an algebraic variety, using for example numerical homotopy methods. We show that the bottleneck degree is a function of classical invariants such as Chern classes and polar classes. We give the formula explicitly in low dimension and provide an algorithm to compute it in the general case.
{"title":"The Bottleneck Degree of Algebraic Varieties","authors":"S. Rocco, David Eklund, Madeleine Weinstein","doi":"10.1137/19m1265776","DOIUrl":"https://doi.org/10.1137/19m1265776","url":null,"abstract":"A bottleneck of a smooth algebraic variety $X subset mathbb{C}^n$ is a pair of distinct points $(x,y) in X$ such that the Euclidean normal spaces at $x$ and $y$ contain the line spanned by $x$ and $y$. The narrowness of bottlenecks is a fundamental complexity measure in the algebraic geometry of data. In this paper we study the number of bottlenecks of affine and projective varieties, which we call the bottleneck degree. The bottleneck degree is a measure of the complexity of computing all bottlenecks of an algebraic variety, using for example numerical homotopy methods. We show that the bottleneck degree is a function of classical invariants such as Chern classes and polar classes. We give the formula explicitly in low dimension and provide an algorithm to compute it in the general case.","PeriodicalId":48489,"journal":{"name":"SIAM Journal on Applied Algebra and Geometry","volume":"42 1","pages":"227-253"},"PeriodicalIF":1.2,"publicationDate":"2019-04-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89357912","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We propose a new way of thinking about one parameter persistence. We believe topological persistence is fundamentally not about decomposition theorems but a central role is played by a choice of metrics. Choosing a pseudometric between persistent vector spaces leads to stabilization of discrete invariants. We develop theory behind this stabilization and stable rank invariant. We give evidence of the usefulness of this approach in concrete data analysis.
{"title":"Metrics and Stabilization in One Parameter Persistence","authors":"W. Chachólski, H. Riihimäki","doi":"10.1137/19m1243932","DOIUrl":"https://doi.org/10.1137/19m1243932","url":null,"abstract":"We propose a new way of thinking about one parameter persistence. We believe topological persistence is fundamentally not about decomposition theorems but a central role is played by a choice of metrics. Choosing a pseudometric between persistent vector spaces leads to stabilization of discrete invariants. We develop theory behind this stabilization and stable rank invariant. We give evidence of the usefulness of this approach in concrete data analysis.","PeriodicalId":48489,"journal":{"name":"SIAM Journal on Applied Algebra and Geometry","volume":"99 1","pages":"69-98"},"PeriodicalIF":1.2,"publicationDate":"2019-04-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84284559","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We prove that the set of singular configurations of a general Gough Stewart platform has a rational parametrization. We introduce a reciprocal twist mapping which, for a general orientation of the platform, realizes the cubic surface of singularities as the blowing up of a quadric surface in five points.
{"title":"Rationality of the Locus of Singularities of the General Gough-Stewart Platform","authors":"M. Coste, Seydou Moussa","doi":"10.1137/19m1253277","DOIUrl":"https://doi.org/10.1137/19m1253277","url":null,"abstract":"We prove that the set of singular configurations of a general Gough Stewart platform has a rational parametrization. We introduce a reciprocal twist mapping which, for a general orientation of the platform, realizes the cubic surface of singularities as the blowing up of a quadric surface in five points.","PeriodicalId":48489,"journal":{"name":"SIAM Journal on Applied Algebra and Geometry","volume":"53 1","pages":"401-421"},"PeriodicalIF":1.2,"publicationDate":"2019-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76909234","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Motivated by numerical integration on manifolds, we relate the algebraic properties of invariant connections to their geometric properties. Using this perspective, we generalize some classical resu...
{"title":"Invariant Connections, Lie Algebra Actions, and Foundations of Numerical Integration on Manifolds","authors":"H. Munthe-Kaas, A. Stern, Olivier Verdier","doi":"10.1137/19M1252879","DOIUrl":"https://doi.org/10.1137/19M1252879","url":null,"abstract":"Motivated by numerical integration on manifolds, we relate the algebraic properties of invariant connections to their geometric properties. Using this perspective, we generalize some classical resu...","PeriodicalId":48489,"journal":{"name":"SIAM Journal on Applied Algebra and Geometry","volume":"4 1","pages":"49-68"},"PeriodicalIF":1.2,"publicationDate":"2019-03-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87371313","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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