Real algebra is usually thought of as the study of certain kinds of preorders on fields and rings. Among its core themes are the separation theorems known as Positivstellens"atze. However, there is a nascent subfield of real algebra which studies preordered semirings and semifields, which is motivated by applications to probability, graph theory and theoretical computer science, among others. Here, we contribute to this subfield by developing a number of foundational results for it, with two abstract Vergleichsstellens"atze being our main theorems. Our first Vergleichsstellensatz states that every semifield preorder is the intersection of its total extensions. We apply this to derive our second main result, a Vergleichsstellensatz for certain non-Archimedean preordered semirings in which the homomorphisms to the tropical reals play an important role. We show how this result recovers the existing Vergleichsstellensatz of Strassen and (through the latter) the classical Positivstellensatz of Krivine--Kadison--Dubois.
{"title":"Abstract Vergleichsstellensätze for Preordered Semifields and Semirings I","authors":"T. Fritz","doi":"10.1137/22M1498413","DOIUrl":"https://doi.org/10.1137/22M1498413","url":null,"abstract":"Real algebra is usually thought of as the study of certain kinds of preorders on fields and rings. Among its core themes are the separation theorems known as Positivstellens\"atze. However, there is a nascent subfield of real algebra which studies preordered semirings and semifields, which is motivated by applications to probability, graph theory and theoretical computer science, among others. Here, we contribute to this subfield by developing a number of foundational results for it, with two abstract Vergleichsstellens\"atze being our main theorems. Our first Vergleichsstellensatz states that every semifield preorder is the intersection of its total extensions. We apply this to derive our second main result, a Vergleichsstellensatz for certain non-Archimedean preordered semirings in which the homomorphisms to the tropical reals play an important role. We show how this result recovers the existing Vergleichsstellensatz of Strassen and (through the latter) the classical Positivstellensatz of Krivine--Kadison--Dubois.","PeriodicalId":48489,"journal":{"name":"SIAM Journal on Applied Algebra and Geometry","volume":"48 1","pages":"505-547"},"PeriodicalIF":1.2,"publicationDate":"2020-03-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90008740","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 characterize the combinatorial types of symmetric frameworks in the plane that are minimally generically symmetry-forced infinitesimally rigid when the underlying symmetry group consists of rotations and translations. Along the way, we use tropical geometry to show how a construction of Edmonds and Rota that associates a matroid to a submodular function can be used to give a description of the algebraic matroid underlying a Hadamard product of two linear spaces in terms of the matroids underlying each linear space. This leads to new, short, proofs of Laman's theorem, and a theorem of Jord{a}n, Kaszanitzky, and Tanigawa characterizing the minimally generically symmetry-forced rigid graphs in the plane when the symmetry group contains only rotations.
{"title":"Generic Symmetry-Forced Infinitesimal Rigidity: Translations and Rotations","authors":"D. Bernstein","doi":"10.1137/20m1346961","DOIUrl":"https://doi.org/10.1137/20m1346961","url":null,"abstract":"We characterize the combinatorial types of symmetric frameworks in the plane that are minimally generically symmetry-forced infinitesimally rigid when the underlying symmetry group consists of rotations and translations. Along the way, we use tropical geometry to show how a construction of Edmonds and Rota that associates a matroid to a submodular function can be used to give a description of the algebraic matroid underlying a Hadamard product of two linear spaces in terms of the matroids underlying each linear space. This leads to new, short, proofs of Laman's theorem, and a theorem of Jord{a}n, Kaszanitzky, and Tanigawa characterizing the minimally generically symmetry-forced rigid graphs in the plane when the symmetry group contains only rotations.","PeriodicalId":48489,"journal":{"name":"SIAM Journal on Applied Algebra and Geometry","volume":"3 1","pages":"190-215"},"PeriodicalIF":1.2,"publicationDate":"2020-03-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87878052","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}
Building on the theory of causal discovery from observational data, we study interactions between multiple (sets of) random variables in a linear structural equation model with non-Gaussian error terms. We give a correspondence between structure in the higher order cumulants and combinatorial structure in the causal graph. It has previously been shown that low rank of the covariance matrix corresponds to trek separation in the graph. Generalizing this criterion to multiple sets of vertices, we characterize when determinants of subtensors of the higher order cumulant tensors vanish. This criterion applies when hidden variables are present as well. For instance, it allows us to identify the presence of a hidden common cause of k of the observed variables.
{"title":"Multi-Trek Separation in Linear Structural Equation Models","authors":"Elina Robeva, Jean-Baptiste Seby","doi":"10.1137/20M1316470","DOIUrl":"https://doi.org/10.1137/20M1316470","url":null,"abstract":"Building on the theory of causal discovery from observational data, we study interactions between multiple (sets of) random variables in a linear structural equation model with non-Gaussian error terms. We give a correspondence between structure in the higher order cumulants and combinatorial structure in the causal graph. It has previously been shown that low rank of the covariance matrix corresponds to trek separation in the graph. Generalizing this criterion to multiple sets of vertices, we characterize when determinants of subtensors of the higher order cumulant tensors vanish. This criterion applies when hidden variables are present as well. For instance, it allows us to identify the presence of a hidden common cause of k of the observed variables.","PeriodicalId":48489,"journal":{"name":"SIAM Journal on Applied Algebra and Geometry","volume":"157 1","pages":"278-303"},"PeriodicalIF":1.2,"publicationDate":"2020-01-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76699715","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}
Given a compact basic semi-algebraic set we provide a numerical scheme to approximate as closely as desired, any finite number of moments of the Hausdorff measure on the boundary of this set. This also allows one to approximate interesting quantities like length, surface, or more general integrals on the boundary, as closely as desired from above and below.
{"title":"Computing the Hausdorff Boundary Measure of Semialgebraic Sets","authors":"J. Lasserre, Victor Magron","doi":"10.1137/20M1314392","DOIUrl":"https://doi.org/10.1137/20M1314392","url":null,"abstract":"Given a compact basic semi-algebraic set we provide a numerical scheme to approximate as closely as desired, any finite number of moments of the Hausdorff measure on the boundary of this set. This also allows one to approximate interesting quantities like length, surface, or more general integrals on the boundary, as closely as desired from above and below.","PeriodicalId":48489,"journal":{"name":"SIAM Journal on Applied Algebra and Geometry","volume":"104 3 Pt 2 1","pages":"441-469"},"PeriodicalIF":1.2,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89684117","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 consider the asymptotic distribution of the IP sparsity function, which measures the minimal support of optimal IP solutions, and the IP to LP distance function, which measures the distance between optimal IP and LP solutions. To this end, we create a framework for studying the asymptotic distribution of general functions related to integer optimization. While there has been a significant amount of research focused around the extreme values that these functions can attain, little is known about their typical values. Each of these functions is defined for a fixed constraint matrix and objective vector while the right hand sides are treated as input. We show that the typical values of these functions are smaller than the known worst case bounds by providing a spectrum of probability-like results that govern their overall asymptotic distributions.
{"title":"The Distributions of Functions Related to Parametric Integer Optimization","authors":"Timm Oertel, Joseph Paat, R. Weismantel","doi":"10.1137/19M1275954","DOIUrl":"https://doi.org/10.1137/19M1275954","url":null,"abstract":"We consider the asymptotic distribution of the IP sparsity function, which measures the minimal support of optimal IP solutions, and the IP to LP distance function, which measures the distance between optimal IP and LP solutions. To this end, we create a framework for studying the asymptotic distribution of general functions related to integer optimization. While there has been a significant amount of research focused around the extreme values that these functions can attain, little is known about their typical values. Each of these functions is defined for a fixed constraint matrix and objective vector while the right hand sides are treated as input. We show that the typical values of these functions are smaller than the known worst case bounds by providing a spectrum of probability-like results that govern their overall asymptotic distributions.","PeriodicalId":48489,"journal":{"name":"SIAM Journal on Applied Algebra and Geometry","volume":"23 1","pages":"422-440"},"PeriodicalIF":1.2,"publicationDate":"2019-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87267415","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 metric space of phylogenetic trees defined by Billera, Holmes, and Vogtmann [Adv. in Appl. Math., 27 (2001), pp. 733--767], which we refer to as BHV space, provides a natural geometric setting ...
Billera, Holmes, and Vogtmann定义的系统发育树的度量空间[j]。数学。， 27(2001)，第733—767页]，我们称之为BHV空间，它提供了一个自然的几何环境……
{"title":"Representations of Partial Leaf Sets in Phylogenetic Tree Space","authors":"Gillian Grindstaff, Megan Owen","doi":"10.1137/18m1235855","DOIUrl":"https://doi.org/10.1137/18m1235855","url":null,"abstract":"The metric space of phylogenetic trees defined by Billera, Holmes, and Vogtmann [Adv. in Appl. Math., 27 (2001), pp. 733--767], which we refer to as BHV space, provides a natural geometric setting ...","PeriodicalId":48489,"journal":{"name":"SIAM Journal on Applied Algebra and Geometry","volume":"58 1","pages":"691-720"},"PeriodicalIF":1.2,"publicationDate":"2019-12-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83655833","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}
SIC-POVMs are configurations of points or rank-one projections arising from the action of a finite Heisenberg group on $mathbb C^d$. The resulting equations are interpreted in terms of moment maps by focussing attention on the orbit of a cyclic subgroup and the maximal torus in $mathrm U(d)$ that contains it. The image of a SIC-POVM under the associated moment map lies in an intersection of real quadrics, which we describe explicitly. We also elaborate the conjectural description of the related number fields and describe the structure of Galois orbits of overlap phases.
sic - povm是由有限Heisenberg群作用于$mathbb C^d$而产生的点或秩一投影的构型。通过将注意力集中在循环子群的轨道和包含它的$ maththrm U(d)$中的最大环面上，用矩映射来解释所得到的方程。在相关矩映射下，SIC-POVM的像位于实二次曲线的交点上，我们对其进行了明确的描述。我们还对相关数场进行了推测描述，并描述了重叠相伽罗瓦轨道的结构。
{"title":"Moment Maps and Galois Orbits in Quantum Information Theory","authors":"Kael Dixon, S. Salamon","doi":"10.1137/19m1305574","DOIUrl":"https://doi.org/10.1137/19m1305574","url":null,"abstract":"SIC-POVMs are configurations of points or rank-one projections arising from the action of a finite Heisenberg group on $mathbb C^d$. The resulting equations are interpreted in terms of moment maps by focussing attention on the orbit of a cyclic subgroup and the maximal torus in $mathrm U(d)$ that contains it. The image of a SIC-POVM under the associated moment map lies in an intersection of real quadrics, which we describe explicitly. We also elaborate the conjectural description of the related number fields and describe the structure of Galois orbits of overlap phases.","PeriodicalId":48489,"journal":{"name":"SIAM Journal on Applied Algebra and Geometry","volume":"1 1","pages":"502-531"},"PeriodicalIF":1.2,"publicationDate":"2019-12-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90046957","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 Whitney numbers of the first kind of combinatorial geometries. The first part of the paper is devoted to general results relating the Mobius functions of nested atomistic lattices, extending some classical theorems in combinatorics. We then specialize our results to restriction geometries, i.e., to sublattices $mathcal{L}(A)$ of the lattice of subspaces of an $mathbb{F}_q$-linear space, say $X$, generated by a set of projective points $A subseteq X$. In this context, we introduce the notion of subspace distribution, and show that partial knowledge of the latter is equivalent to partial knowledge of the Whitney numbers of $mathcal{L}(A)$. This refines a classical result by Dowling. The most interesting applications of our results are to be seen in the theory of higher-weight Dowling lattices (HWDLs), to which we dovote the second and most substantive part of the paper. These combinatorial geometries were introduced by Dowling in 1971 in connection with fundamental problems in coding theory, and further studied, among others, by Zaslavsky, Bonin, Kung, Brini, and Games. To date, still very little is known about these lattices. In particular, the techniques to compute their Whitney numbers have not been discovered yet. In this paper, we bring forward the theory of HWDLs, computing their Whitney numbers for new infinite families of parameters. Moreover, we show that the second Whitney numbers of HWDLs are polynomials in the underlying field size $q$, whose coefficients are expressions involving the Bernoulli numbers. This reveals a new link between combinatorics, coding theory, and number theory. We also study the asymptotics of the Whitney numbers of HWDLs as the field size grows, giving upper bounds and exact estimates in some cases. In passing, we obtain new results on the density functions of error-correcting codes.
{"title":"Whitney Numbers of Combinatorial Geometries and Higher-Weight Dowling Lattices","authors":"A. Ravagnani","doi":"10.1137/20m1382635","DOIUrl":"https://doi.org/10.1137/20m1382635","url":null,"abstract":"We study the Whitney numbers of the first kind of combinatorial geometries. The first part of the paper is devoted to general results relating the Mobius functions of nested atomistic lattices, extending some classical theorems in combinatorics. We then specialize our results to restriction geometries, i.e., to sublattices $mathcal{L}(A)$ of the lattice of subspaces of an $mathbb{F}_q$-linear space, say $X$, generated by a set of projective points $A subseteq X$. In this context, we introduce the notion of subspace distribution, and show that partial knowledge of the latter is equivalent to partial knowledge of the Whitney numbers of $mathcal{L}(A)$. This refines a classical result by Dowling. \u0000The most interesting applications of our results are to be seen in the theory of higher-weight Dowling lattices (HWDLs), to which we dovote the second and most substantive part of the paper. These combinatorial geometries were introduced by Dowling in 1971 in connection with fundamental problems in coding theory, and further studied, among others, by Zaslavsky, Bonin, Kung, Brini, and Games. To date, still very little is known about these lattices. In particular, the techniques to compute their Whitney numbers have not been discovered yet. In this paper, we bring forward the theory of HWDLs, computing their Whitney numbers for new infinite families of parameters. Moreover, we show that the second Whitney numbers of HWDLs are polynomials in the underlying field size $q$, whose coefficients are expressions involving the Bernoulli numbers. This reveals a new link between combinatorics, coding theory, and number theory. We also study the asymptotics of the Whitney numbers of HWDLs as the field size grows, giving upper bounds and exact estimates in some cases. In passing, we obtain new results on the density functions of error-correcting codes.","PeriodicalId":48489,"journal":{"name":"SIAM Journal on Applied Algebra and Geometry","volume":"66 1","pages":"156-189"},"PeriodicalIF":1.2,"publicationDate":"2019-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72788425","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 convex form of degree larger than one is always nonnegative since it vanishes together with its gradient at the origin. In 2007, Parrilo asked if convex forms are always sums of squares. A few years later, Blekherman answered the question in the negative by showing through volume arguments that for high enough number of variables, there must be convex forms of degree as low as 4 that are not sums of squares. Remarkably, no examples are known to date. In this paper, we show that all convex forms in 4 variables and of degree 4 are sums of squares. We also show that if a conjecture of Blekherman related to the so-called Cayley-Bacharach relations is true, then the same statement holds for convex forms in 3 variables and of degree 6. These are the two minimal cases where one would have any hope of seeing convex forms that are not sums of squares (due to known obstructions). A main ingredient of the proof is the derivation of certain "generalized Cauchy-Schwarz inequalities" which could be of independent interest.
{"title":"On Sum of Squares Representation of Convex Forms and Generalized Cauchy-Schwarz Inequalities","authors":"Bachir El Khadir","doi":"10.1137/19m1287584","DOIUrl":"https://doi.org/10.1137/19m1287584","url":null,"abstract":"A convex form of degree larger than one is always nonnegative since it vanishes together with its gradient at the origin. In 2007, Parrilo asked if convex forms are always sums of squares. A few years later, Blekherman answered the question in the negative by showing through volume arguments that for high enough number of variables, there must be convex forms of degree as low as 4 that are not sums of squares. Remarkably, no examples are known to date. In this paper, we show that all convex forms in 4 variables and of degree 4 are sums of squares. We also show that if a conjecture of Blekherman related to the so-called Cayley-Bacharach relations is true, then the same statement holds for convex forms in 3 variables and of degree 6. These are the two minimal cases where one would have any hope of seeing convex forms that are not sums of squares (due to known obstructions). A main ingredient of the proof is the derivation of certain \"generalized Cauchy-Schwarz inequalities\" which could be of independent interest.","PeriodicalId":48489,"journal":{"name":"SIAM Journal on Applied Algebra and Geometry","volume":"21 4 1","pages":"377-400"},"PeriodicalIF":1.2,"publicationDate":"2019-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89655844","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 consider the problem of finding the smallest rank of a complex matrix whose absolute values of the entries are given. We call this minimum the phaseless rank of the matrix of the entrywise absolute values. In this paper we study this quantity, extending a classic result of Camion and Hoffman and connecting it to the study of amoebas of determinantal varieties and of semidefinite representations of convex sets. As a consequence, we prove that the set of maximal minors of a matrix of indeterminates form an amoeba basis for the ideal they define, and we attain a new upper bound on the complex semidefinite extension complexity of polytopes, dependent only on their number of vertices and facets. We also highlight the connections between the notion of phaseless rank and the problem of finding large sets of complex equiangular lines or mutually unbiased bases.
{"title":"The Phaseless Rank of a Matrix","authors":"António Pedro Goucha, J. Gouveia","doi":"10.1137/19M1289820","DOIUrl":"https://doi.org/10.1137/19M1289820","url":null,"abstract":"We consider the problem of finding the smallest rank of a complex matrix whose absolute values of the entries are given. We call this minimum the phaseless rank of the matrix of the entrywise absolute values. In this paper we study this quantity, extending a classic result of Camion and Hoffman and connecting it to the study of amoebas of determinantal varieties and of semidefinite representations of convex sets. As a consequence, we prove that the set of maximal minors of a matrix of indeterminates form an amoeba basis for the ideal they define, and we attain a new upper bound on the complex semidefinite extension complexity of polytopes, dependent only on their number of vertices and facets. We also highlight the connections between the notion of phaseless rank and the problem of finding large sets of complex equiangular lines or mutually unbiased bases.","PeriodicalId":48489,"journal":{"name":"SIAM Journal on Applied Algebra and Geometry","volume":"22 1","pages":"526-551"},"PeriodicalIF":1.2,"publicationDate":"2019-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88110943","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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