{"title":"How many zeros of a random sparse polynomial are real?","authors":"Gorav Jindal, Anurag Pandey, Himanshu Shukla, Charilaos Zisopoulos","doi":"10.1145/3373207.3404031","DOIUrl":"https://doi.org/10.1145/3373207.3404031","url":null,"abstract":"We investigate the number of real zeros of a univariate k-sparse polynomial f over the reals, when the coefficients of f come from independent standard normal distributions. Recently Bürgisser, Ergür and Tonelli-Cueto showed that the expected number of real zeros of f in such cases is bounded by [EQUATION]. In this work, we improve the bound to [EQUATION] and also show that this bound is tight by constructing a family of sparse support whose expected number of real zeros is lower bounded by [EQUATION]. Our main technique is an alternative formulation of the Kac integral by Edelman-Kostlan which allows us to bound the expected number of zeros of f in terms of the expected number of zeros of polynomials of lower sparsity. Using our technique, we also recover the O (log n) bound on the expected number of real zeros of a dense polynomial of degree n with coefficients coming from independent standard normal distributions.","PeriodicalId":186699,"journal":{"name":"Proceedings of the 45th International Symposium on Symbolic and Algebraic Computation","volume":"19 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124457688","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Decidability of membership problems for flat rational subsets of GL(2, Q) and singular matrices","authors":"V. Diekert, I. Potapov, P. Semukhin","doi":"10.1145/3373207.3404038","DOIUrl":"https://doi.org/10.1145/3373207.3404038","url":null,"abstract":"This work relates numerical problems on matrices over the rationals to symbolic algorithms on words and finite automata. Using exact algebraic algorithms and symbolic computation, we prove new decidability results for 2 × 2 matrices over Q. Namely, we introduce a notion of flat rational sets: if M is a monoid and N ≤ M is its submonoid, then flat rational sets of M relative to N are finite unions of the form L0g1 L1 ··· gtLt where all Lis are rational subsets of N and gi ∈ M. We give quite general sufficient conditions under which flat rational sets form an effective relative Boolean algebra. As a corollary, we obtain that the emptiness problem for Boolean combinations of flat rational subsets of GL(2, Q) over GL(2, Z) is decidable. We also show a dichotomy for nontrivial group extension of GL(2, Z) in GL(2, Q): if G is a f.g. group such that GL(2, Z) < G ≤ GL(2, Q), then either G ≅ GL(2, Z) × Zk, for some k ≥ 1, or G contains an extension of the Baumslag-Solitar group BS(1, q), with q ≥ 2, of infinite index. It turns out that in the first case the membership problem for G is decidable but the equality problem for rational subsets of G is undecidable. In the second case, decidability of the membership problem is open for every such G. In the last section we prove new decidability results for flat rational sets that contain singular matrices. In particular, we show that the membership problem is decidable for flat rational subsets of M(2, Q) relative to the submonoid that is generated by the matrices from M(2, Z) with determinants 0, ± 1 and the central rational matrices.","PeriodicalId":186699,"journal":{"name":"Proceedings of the 45th International Symposium on Symbolic and Algebraic Computation","volume":"150 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-10-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124198599","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"A second order cone characterization for sums of nonnegative circuits","authors":"Jie Wang, Victor Magron","doi":"10.1145/3373207.3404033","DOIUrl":"https://doi.org/10.1145/3373207.3404033","url":null,"abstract":"The second-order cone (SOC) is a class of simple convex cones and optimizing over them can be done more efficiently than with semidefinite programming. It is interesting both in theory and in practice to investigate which convex cones admit a representation using SOCs, given that they have a strong expressive ability. In this paper, we prove constructively that the cone of sums of nonnegative circuits (SONC) admits an SOC representation. Based on this, we give a new algorithm to compute SONC decompositions for certain classes of nonnegative polynomials via SOC programming. Numerical experiments demonstrate the efficiency of our algorithm for polynomials with a fairly large size (both size of degree and number of variables).","PeriodicalId":186699,"journal":{"name":"Proceedings of the 45th International Symposium on Symbolic and Algebraic Computation","volume":"480 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-06-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133579171","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}