View in EDS HTML Full Text PDF Full Text

Sums of Squares Certificates for Polynomial Moment Inequalities.

Saved in:
Bibliographic Details
Title: Sums of Squares Certificates for Polynomial Moment Inequalities.
Authors: Klep, Igor1,2,3 (AUTHOR) igor.klep@fmf.uni-lj.si, Magron, Victor4 (AUTHOR) vmagron@laas.fr, Volčič, Jurij5 (AUTHOR) jurij.volcic@auckland.ac.nz
Source: Foundations of Computational Mathematics. Jun2026, Vol. 26 Issue 3, p1381-1423. 43p.
Subjects: Sum of squares, Semialgebraic sets, Polynomials, Bell's theorem, Hilbert, David, 1862-1943, Quantum theory, Semidefinite programming
Abstract: This paper introduces and develops the algebraic framework of moment polynomials, which are polynomial expressions in commuting variables and their formal mixed moments. Their positivity and optimization over probability measures supported on semialgebraic sets and subject to moment polynomial constraints is investigated. On the one hand, a positive solution to Hilbert's 17th problem for pseudo-moments is given. On the other hand, moment polynomials positive on actual measures are shown to be sums of squares and formal moments of squares up to arbitrarily small perturbation of their coefficients. When only measures supported on a bounded semialgebraic set are considered, a stronger algebraic certificate for moment polynomial positivity is derived. This result gives rise to a converging hierarchy of semidefinite programs for moment polynomial optimization. Finally, as an application, two open nonlinear Bell inequalities from quantum physics are settled. [ABSTRACT FROM AUTHOR]
Copyright of Foundations of Computational Mathematics is the property of Springer Nature and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
Database: Engineering Source
Full text is not displayed to guests.
FullText Links:
  – Type: pdflink
Text:
  Availability: 1
Header DbId: egs
DbLabel: Engineering Source
An: 194201081
AccessLevel: 6
PubType: Academic Journal
PubTypeId: academicJournal
PreciseRelevancyScore: 0
IllustrationInfo
Items – Name: Title
  Label: Title
  Group: Ti
  Data: Sums of Squares Certificates for Polynomial Moment Inequalities.
– Name: Author
  Label: Authors
  Group: Au
  Data: <searchLink fieldCode="AR" term="%22Klep%2C+Igor%22">Klep, Igor</searchLink><relatesTo>1,2,3</relatesTo> (AUTHOR)<i> igor.klep@fmf.uni-lj.si</i><br /><searchLink fieldCode="AR" term="%22Magron%2C+Victor%22">Magron, Victor</searchLink><relatesTo>4</relatesTo> (AUTHOR)<i> vmagron@laas.fr</i><br /><searchLink fieldCode="AR" term="%22Volčič%2C+Jurij%22">Volčič, Jurij</searchLink><relatesTo>5</relatesTo> (AUTHOR)<i> jurij.volcic@auckland.ac.nz</i>
– Name: TitleSource
  Label: Source
  Group: Src
  Data: <searchLink fieldCode="JN" term="%22Foundations+of+Computational+Mathematics%22">Foundations of Computational Mathematics</searchLink>. Jun2026, Vol. 26 Issue 3, p1381-1423. 43p.
– Name: Subject
  Label: Subjects
  Group: Su
  Data: <searchLink fieldCode="DE" term="%22Sum+of+squares%22">Sum of squares</searchLink><br /><searchLink fieldCode="DE" term="%22Semialgebraic+sets%22">Semialgebraic sets</searchLink><br /><searchLink fieldCode="DE" term="%22Polynomials%22">Polynomials</searchLink><br /><searchLink fieldCode="DE" term="%22Bell's+theorem%22">Bell's theorem</searchLink><br /><searchLink fieldCode="DE" term="%22Hilbert%2C+David%2C+1862-1943%22">Hilbert, David, 1862-1943</searchLink><br /><searchLink fieldCode="DE" term="%22Quantum+theory%22">Quantum theory</searchLink><br /><searchLink fieldCode="DE" term="%22Semidefinite+programming%22">Semidefinite programming</searchLink>
– Name: Abstract
  Label: Abstract
  Group: Ab
  Data: This paper introduces and develops the algebraic framework of moment polynomials, which are polynomial expressions in commuting variables and their formal mixed moments. Their positivity and optimization over probability measures supported on semialgebraic sets and subject to moment polynomial constraints is investigated. On the one hand, a positive solution to Hilbert's 17th problem for pseudo-moments is given. On the other hand, moment polynomials positive on actual measures are shown to be sums of squares and formal moments of squares up to arbitrarily small perturbation of their coefficients. When only measures supported on a bounded semialgebraic set are considered, a stronger algebraic certificate for moment polynomial positivity is derived. This result gives rise to a converging hierarchy of semidefinite programs for moment polynomial optimization. Finally, as an application, two open nonlinear Bell inequalities from quantum physics are settled. [ABSTRACT FROM AUTHOR]
– Name: AbstractSuppliedCopyright
  Label:
  Group: Ab
  Data: <i>Copyright of Foundations of Computational Mathematics is the property of Springer Nature and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.)
PLink https://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=egs&AN=194201081
RecordInfo BibRecord:
  BibEntity:
    Identifiers:
      – Type: doi
        Value: 10.1007/s10208-025-09703-z
    Languages:
      – Code: eng
        Text: English
    PhysicalDescription:
      Pagination:
        PageCount: 43
        StartPage: 1381
    Subjects:
      – SubjectFull: Sum of squares
        Type: general
      – SubjectFull: Semialgebraic sets
        Type: general
      – SubjectFull: Polynomials
        Type: general
      – SubjectFull: Bell's theorem
        Type: general
      – SubjectFull: Hilbert, David, 1862-1943
        Type: general
      – SubjectFull: Quantum theory
        Type: general
      – SubjectFull: Semidefinite programming
        Type: general
    Titles:
      – TitleFull: Sums of Squares Certificates for Polynomial Moment Inequalities.
        Type: main
  BibRelationships:
    HasContributorRelationships:
      – PersonEntity:
          Name:
            NameFull: Klep, Igor
      – PersonEntity:
          Name:
            NameFull: Magron, Victor
      – PersonEntity:
          Name:
            NameFull: Volčič, Jurij
    IsPartOfRelationships:
      – BibEntity:
          Dates:
            – D: 01
              M: 06
              Text: Jun2026
              Type: published
              Y: 2026
          Identifiers:
            – Type: issn-print
              Value: 16153375
          Numbering:
            – Type: volume
              Value: 26
            – Type: issue
              Value: 3
          Titles:
            – TitleFull: Foundations of Computational Mathematics
              Type: main
ResultId 1