Sandra Di Rocco, Parker B. Edwards, David Eklund, Oliver Gäfvert, Jonathan D. Hauenstein
{"title":"Computing Geometric Feature Sizes for Algebraic Manifolds","authors":"Sandra Di Rocco, Parker B. Edwards, David Eklund, Oliver Gäfvert, Jonathan D. Hauenstein","doi":"10.1137/22m1522656","DOIUrl":"https://doi.org/10.1137/22m1522656","url":null,"abstract":"","PeriodicalId":48489,"journal":{"name":"SIAM Journal on Applied Algebra and Geometry","volume":" 34","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135243099","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 present an algebraic characterization of perfect graphs, i.e., graphs for which the clique number and the chromatic number coincide for every induced subgraph. We show that a graph is perfect if and only if certain nonnegative polynomials associated with the graph are sums of squares. As a byproduct, we obtain several infinite families of nonnegative polynomials that are not sums of squares through graph-theoretic constructions. We also characterize graphs for which the associated polynomials belong to certain structured subsets of sum of squares polynomials. Finally, we reformulate some well-known results from the theory of perfect graphs as statements about sum of squares proofs of nonnegativity of certain polynomials.
{"title":"A Sum of Squares Characterization of Perfect Graphs","authors":"Amir Ali Ahmadi, Cemil Dibek","doi":"10.1137/22m1530410","DOIUrl":"https://doi.org/10.1137/22m1530410","url":null,"abstract":"We present an algebraic characterization of perfect graphs, i.e., graphs for which the clique number and the chromatic number coincide for every induced subgraph. We show that a graph is perfect if and only if certain nonnegative polynomials associated with the graph are sums of squares. As a byproduct, we obtain several infinite families of nonnegative polynomials that are not sums of squares through graph-theoretic constructions. We also characterize graphs for which the associated polynomials belong to certain structured subsets of sum of squares polynomials. Finally, we reformulate some well-known results from the theory of perfect graphs as statements about sum of squares proofs of nonnegativity of certain polynomials.","PeriodicalId":48489,"journal":{"name":"SIAM Journal on Applied Algebra and Geometry","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136077517","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 give an algorithm with singly exponential complexity for computing the barcodes up to dimension (for any fixed ) of the filtration of a given semialgebraic set by the sublevel sets of a given polynomial. Our algorithm is the first algorithm for this problem with singly exponential complexity and generalizes the corresponding results for computing the Betti numbers up to dimension of semialgebraic sets with no filtration present.
{"title":"Persistent Homology of Semialgebraic Sets","authors":"Saugata Basu, Negin Karisani","doi":"10.1137/22m1494415","DOIUrl":"https://doi.org/10.1137/22m1494415","url":null,"abstract":"We give an algorithm with singly exponential complexity for computing the barcodes up to dimension (for any fixed ) of the filtration of a given semialgebraic set by the sublevel sets of a given polynomial. Our algorithm is the first algorithm for this problem with singly exponential complexity and generalizes the corresponding results for computing the Betti numbers up to dimension of semialgebraic sets with no filtration present.","PeriodicalId":48489,"journal":{"name":"SIAM Journal on Applied Algebra and Geometry","volume":"26 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135579655","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}
{"title":"Finiteness of Spatial Central Configurations with Fixed Subconfigurations","authors":"Yiyang Deng, M. Hampton","doi":"10.1137/22m1490788","DOIUrl":"https://doi.org/10.1137/22m1490788","url":null,"abstract":"","PeriodicalId":48489,"journal":{"name":"SIAM Journal on Applied Algebra and Geometry","volume":"19 1","pages":"610-622"},"PeriodicalIF":1.2,"publicationDate":"2023-08-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88211447","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}
{"title":"The Projectivization Matroid of a (boldsymbol{q}) -Matroid","authors":"Benjamin Jany","doi":"10.1137/22m1494567","DOIUrl":"https://doi.org/10.1137/22m1494567","url":null,"abstract":"","PeriodicalId":48489,"journal":{"name":"SIAM Journal on Applied Algebra and Geometry","volume":"7 1","pages":"386-413"},"PeriodicalIF":1.2,"publicationDate":"2023-05-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79795384","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 present an algorithm for computing circuit polynomials in the algebraic rigidity matroid associated to the Cayley–Menger ideal for points in 2D. It relies on combinatorial resultants, a new operation on graphs that captures properties of the Sylvester resultant of two polynomials in this ideal. We show that every rigidity circuit has a construction tree from graphs based on this operation. Our algorithm performs an algebraic elimination guided by such a construction tree and uses classical resultants, factorization, and ideal membership. To highlight its effectiveness, we implemented the algorithm in Mathematica: it took less than 15 seconds on an example where a Gröbner basis calculation took 5 days and 6 hours. Additional speed-ups are obtained using non- generators of the Cayley–Menger ideal and simple variations on our main algorithm.
{"title":"Computing Circuit Polynomials in the Algebraic Rigidity Matroid","authors":"Goran Malić, Ileana Streinu","doi":"10.1137/21m1437986","DOIUrl":"https://doi.org/10.1137/21m1437986","url":null,"abstract":"We present an algorithm for computing circuit polynomials in the algebraic rigidity matroid associated to the Cayley–Menger ideal for points in 2D. It relies on combinatorial resultants, a new operation on graphs that captures properties of the Sylvester resultant of two polynomials in this ideal. We show that every rigidity circuit has a construction tree from graphs based on this operation. Our algorithm performs an algebraic elimination guided by such a construction tree and uses classical resultants, factorization, and ideal membership. To highlight its effectiveness, we implemented the algorithm in Mathematica: it took less than 15 seconds on an example where a Gröbner basis calculation took 5 days and 6 hours. Additional speed-ups are obtained using non- generators of the Cayley–Menger ideal and simple variations on our main algorithm.","PeriodicalId":48489,"journal":{"name":"SIAM Journal on Applied Algebra and Geometry","volume":"24 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-05-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135140263","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}
{"title":"Optimal Encodings to Elliptic Curves of (boldsymbol{j})-Invariants 0, 1728","authors":"D. Koshelev","doi":"10.1137/21m1441602","DOIUrl":"https://doi.org/10.1137/21m1441602","url":null,"abstract":"","PeriodicalId":48489,"journal":{"name":"SIAM Journal on Applied Algebra and Geometry","volume":"12 1","pages":"600-617"},"PeriodicalIF":1.2,"publicationDate":"2022-12-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81751427","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 dissertation, we view matrix frames as representations of quivers and study them within the general framework of Quiver Invariant Theory. We are particularly interested in radial isotropic and Parseval matrix frames. Using methods from Quiver Invariant Theory [CD21], we first prove a far-reaching generalization of Barthe's Theorem [Bar98] on vectors in radial isotropic position to the case of matrix frames (see Theorems 5.13(3) and 4.12). With this tool at our disposal, we generalize the Paulsen problem from frames (of vectors) to frames of matrices of arbitrary rank and size extending Hamilton-Moitra's upper bound [HM18]. Specifically, we show in Theorem 5.20 that for any given ε-nearly equal-norm Parseval frame F of n matrices with d rows there exists an equal-norm Parseval frame W of n matrices with d rows such that dist^2 (F,W) [less than or equal to] 46[epsilon]d^2. Finally, in Theorem 5.28 we address the constructive aspects of transforming a matrix frame into radial isotropic position which extend those in [Bar98, AKS20].
本文将矩阵框架视为颤振的表示，并在颤振不变性理论的一般框架内对其进行了研究。我们对径向各向同性和Parseval矩阵框架特别感兴趣。利用Quiver Invariant Theory [CD21]中的方法，我们首先证明了Barthe定理[Bar98]在径向各向同性位置上对矩阵框架的推广(见定理5.13(3)和4.12)。利用这个工具，我们将Paulsen问题从(向量的)框架推广到扩展Hamilton-Moitra上界的任意秩和大小的矩阵框架[HM18]。具体地，我们在定理5.20中证明了对于任意给定的ε-近似等范数Parseval坐标系F (n个d行矩阵)存在一个包含n个d行矩阵的等范数Parseval坐标系W，使得dist^2 (F,W)[小于或等于]46[ε]d^2。最后，在定理5.28中，我们讨论了将矩阵框架转换为径向各向同性位置的建设性方面，扩展了[Bar98, AKS20]中的内容。
{"title":"A Quiver Invariant Theoretic Approach to Radial Isotropy and the Paulsen Problem for Matrix Frames","authors":"C. Chindris, Jasim Ismaeel","doi":"10.1137/21m141470x","DOIUrl":"https://doi.org/10.1137/21m141470x","url":null,"abstract":"In this dissertation, we view matrix frames as representations of quivers and study them within the general framework of Quiver Invariant Theory. We are particularly interested in radial isotropic and Parseval matrix frames. Using methods from Quiver Invariant Theory [CD21], we first prove a far-reaching generalization of Barthe's Theorem [Bar98] on vectors in radial isotropic position to the case of matrix frames (see Theorems 5.13(3) and 4.12). With this tool at our disposal, we generalize the Paulsen problem from frames (of vectors) to frames of matrices of arbitrary rank and size extending Hamilton-Moitra's upper bound [HM18]. Specifically, we show in Theorem 5.20 that for any given ε-nearly equal-norm Parseval frame F of n matrices with d rows there exists an equal-norm Parseval frame W of n matrices with d rows such that dist^2 (F,W) [less than or equal to] 46[epsilon]d^2. Finally, in Theorem 5.28 we address the constructive aspects of transforming a matrix frame into radial isotropic position which extend those in [Bar98, AKS20].","PeriodicalId":48489,"journal":{"name":"SIAM Journal on Applied Algebra and Geometry","volume":"85 1","pages":"536-562"},"PeriodicalIF":1.2,"publicationDate":"2022-11-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75010828","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}
G. Coutinho, P. Baptista, C. Godsil, Thomás Jung Spier, R. Werner
The adjacency matrix of a graph G is the Hamiltonian for a continuous-time quantum walk on the vertices of G. Although the entries of the adjacency matrix are integers, its eigenvalues are generally irrational and, because of this, the behaviour of the walk is typically not periodic. In consequence we can usually only compute numerical approximations to parameters of the walk. In this paper, we develop theory to exactly study any quantum walk generated by an integral Hamiltonian. As a result, we provide exact methods to compute the average of the mixing matrices, and to decide whether pretty good (or almost) perfect state transfer occurs in a given graph. We also use our methods to study geometric properties of beautiful curves arising from entries of the quantum walk matrix, and discuss possible applications of these results.
{"title":"Irrational Quantum Walks","authors":"G. Coutinho, P. Baptista, C. Godsil, Thomás Jung Spier, R. Werner","doi":"10.1137/22m1521262","DOIUrl":"https://doi.org/10.1137/22m1521262","url":null,"abstract":"The adjacency matrix of a graph G is the Hamiltonian for a continuous-time quantum walk on the vertices of G. Although the entries of the adjacency matrix are integers, its eigenvalues are generally irrational and, because of this, the behaviour of the walk is typically not periodic. In consequence we can usually only compute numerical approximations to parameters of the walk. In this paper, we develop theory to exactly study any quantum walk generated by an integral Hamiltonian. As a result, we provide exact methods to compute the average of the mixing matrices, and to decide whether pretty good (or almost) perfect state transfer occurs in a given graph. We also use our methods to study geometric properties of beautiful curves arising from entries of the quantum walk matrix, and discuss possible applications of these results.","PeriodicalId":48489,"journal":{"name":"SIAM Journal on Applied Algebra and Geometry","volume":"13 1","pages":"567-584"},"PeriodicalIF":1.2,"publicationDate":"2022-08-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79008405","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 study Bayesian networks from a commutative algebra perspective. We characterize a class of toric Bayesian nets, and provide the first example of a Bayesian net which is proved non-toric under any linear change of variables. Concerning the class of toric Bayesian nets, we study their quadratic relations and prove a conjecture by Garcia, Stillman, and Sturmfels for this class. In addition, we give a necessary condition on the underlying directed acyclic graph for when all relations are quadratic.
{"title":"Toric and Non-toric Bayesian Networks","authors":"Lisa Nicklasson","doi":"10.1137/22M1515690","DOIUrl":"https://doi.org/10.1137/22M1515690","url":null,"abstract":"In this paper we study Bayesian networks from a commutative algebra perspective. We characterize a class of toric Bayesian nets, and provide the first example of a Bayesian net which is proved non-toric under any linear change of variables. Concerning the class of toric Bayesian nets, we study their quadratic relations and prove a conjecture by Garcia, Stillman, and Sturmfels for this class. In addition, we give a necessary condition on the underlying directed acyclic graph for when all relations are quadratic.","PeriodicalId":48489,"journal":{"name":"SIAM Journal on Applied Algebra and Geometry","volume":"13 1","pages":"549-566"},"PeriodicalIF":1.2,"publicationDate":"2022-08-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74094068","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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