Implicitisation and Parameterisation in Polynomial Functors
In earlier work, the second author showed that a closed subset of a polynomial functor can always be defined by finitely many polynomial equations. In follow-up work on GL ∞ -varieties, Bik–Draisma–Eggermont–Snowden showed, among other things, that in characteristic zero every such closed subset is...
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| Published in: | Foundations of computational mathematics Vol. 24; no. 5; pp. 1567 - 1593 |
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| Abstract | In earlier work, the second author showed that a closed subset of a polynomial functor can always be defined by finitely many polynomial equations. In follow-up work on
GL
∞
-varieties, Bik–Draisma–Eggermont–Snowden showed, among other things, that in characteristic zero every such closed subset is the image of a morphism whose domain is the product of a finite-dimensional affine variety and a polynomial functor. In this paper, we show that both results can be made algorithmic: there exists an algorithm
implicitise
that takes as input a morphism into a polynomial functor and outputs finitely many equations defining the closure of the image; and an algorithm
parameterise
that takes as input a finite set of equations defining a closed subset of a polynomial functor and outputs a morphism whose image is that closed subset. |
|---|---|
| AbstractList | In earlier work, the second author showed that a closed subset of a polynomial functor can always be defined by finitely many polynomial equations. In follow-up work on GL∞-varieties, Bik–Draisma–Eggermont–Snowden showed, among other things, that in characteristic zero every such closed subset is the image of a morphism whose domain is the product of a finite-dimensional affine variety and a polynomial functor. In this paper, we show that both results can be made algorithmic: there exists an algorithm implicitise that takes as input a morphism into a polynomial functor and outputs finitely many equations defining the closure of the image; and an algorithm parameterise that takes as input a finite set of equations defining a closed subset of a polynomial functor and outputs a morphism whose image is that closed subset. In earlier work, the second author showed that a closed subset of a polynomial functor can always be defined by finitely many polynomial equations. In follow-up work on $${\text {GL}}_\infty $$ GL ∞ -varieties, Bik–Draisma–Eggermont–Snowden showed, among other things, that in characteristic zero every such closed subset is the image of a morphism whose domain is the product of a finite-dimensional affine variety and a polynomial functor. In this paper, we show that both results can be made algorithmic: there exists an algorithm $$\textbf{implicitise}$$ implicitise that takes as input a morphism into a polynomial functor and outputs finitely many equations defining the closure of the image; and an algorithm $$\textbf{parameterise}$$ parameterise that takes as input a finite set of equations defining a closed subset of a polynomial functor and outputs a morphism whose image is that closed subset. In earlier work, the second author showed that a closed subset of a polynomial functor can always be defined by finitely many polynomial equations. In follow-up work on GL ∞ -varieties, Bik–Draisma–Eggermont–Snowden showed, among other things, that in characteristic zero every such closed subset is the image of a morphism whose domain is the product of a finite-dimensional affine variety and a polynomial functor. In this paper, we show that both results can be made algorithmic: there exists an algorithm implicitise that takes as input a morphism into a polynomial functor and outputs finitely many equations defining the closure of the image; and an algorithm parameterise that takes as input a finite set of equations defining a closed subset of a polynomial functor and outputs a morphism whose image is that closed subset. |
| Author | Ventura, Emanuele Blatter, Andreas Draisma, Jan |
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| Cites_doi | 10.1016/j.aim.2011.08.009 10.2140/ant.2017.11.2197 10.1090/jams/923 10.1007/s002220050119 10.1007/3-540-54522-0_108. 10.1007/s00222-019-00889-y 10.1007/978-3-319-16721-3. 10.1090/S0025-5718-2010-02415-9 10.4310/MRL.2013.v20.n4.a10 10.1007/s10231-011-0238-6 10.5427/jsing.2018.17g 10.1007/3-540-18170-9_156 10.1007/BF02684802. |
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| References | J.M. Landsberg and Giorgio Ottaviani. Equations for secant varieties of Veronese and other varieties. Ann. Mat. Pura Appl. (4), 192(4):569–606, 2013. DraismaJanTopological Noetherianity of polynomial functorsJ. Am. Math. Soc.201932691707398198610.1090/jams/923 M. J. Greenberg. Rational points in Henselian discrete valuation rings. Publ. Math., Inst. Hautes Étud. Sci., 31:563–568, 1966. https://doi.org/10.1007/BF02684802. David A. Cox, John Little, and Donal O’Shea. Ideals, varieties, and algorithms. Undergraduate Texts in Mathematics. Springer, Cham, fourth edition, 2015. An introduction to computational algebraic geometry and commutative algebra. https://doi.org/10.1007/978-3-319-16721-3. Arthur Bik. Strength and noetherianity for infinite tensors. PhD thesis, Universität Bern, 2020. David A. Cox, John Little, and Donal O’Shea. Using algebraic geometry, volume 185 of Graduate Texts in Mathematics. Springer, New York, second edition, 2005. Toby Ord. How to simulate everything (all at once). URL: http://www.amirrorclear.net/academic/ideas/simulation/index.html. Arthur Bik, Jan Draisma, Rob H. Eggermont, and Andrew Snowden. Uniformity for limits of tensors. 2023. Preprint, arXiv:2305.19866. ErmanDanielSamSteven VSnowdenAndrewBig polynomial rings and Stillman’s conjectureInvent. Math.20192182413439401170310.1007/s00222-019-00889-y Gert-Martin Greuel and Gerhard Pfister. A Singular introduction to commutative algebra. Springer, Berlin, extended edition, 2008. With contributions by Olaf Bachmann, Christoph Lossen and Hans Schönemann. Teresa Krick and Alessandro Logar. An algorithm for the computation of the radical of an ideal in the ring of polynomials. In Applied algebra, algebraic algorithms and error-correcting codes (New Orleans, LA, 1991), volume 539 of Lecture Notes in Comput. Sci., pages 195–205. Springer, Berlin, 1991. https://doi.org/10.1007/3-540-54522-0_108. Arthur Bik, Jan Draisma, Rob H. Eggermont, and Andrew Snowden. The geometry of polynomial representations. Int. Math. Res. Not., 2021. To appear, arXiv:2105.12621. ProcesiClaudioLie Groups. An Approach through Invariants and Representations2007New YorkSpringer Christopher J. Hillar, Robert Krone, and Anton Leykin. Equivariant Gröbner bases. In The 50th anniversary of Gröbner bases. Proceedings of the 8th Mathematical Society of Japan-Seasonal Institute (MSJ-SI 2015), Osaka, Japan, July 1–10, 2015, pages 129–154. Tokyo: Mathematical Society of Japan (MSJ), 2018. William Fulton and Joe Harris. Representation Theory. A First Course. Number 129 in Graduate Texts in Mathematics. Springer-Verlag, New York, 1991. Daniel E. Cohen. Closure relations, Buchberger’s algorithm, and polynomials in infinitely many variables. In Computation theory and logic, volume 270 of Lect. Notes Comput. Sci., pages 78–87, 1987. FriedlanderEric MSuslinAndreiCohomology of finite group schemes over a fieldInvent. Math.19971272209270142761810.1007/s002220050119 Christopher J. Hillar, Robert Krone, and Anton Leykin. EquivariantGB Macaulay2 package. Webpage: http://people.math.gatech.edu/~rkrone3/EquivariantGB.html, 2013. HillarChristopher JSullivantSethFinite Gröbner bases in infinite dimensional polynomial rings and applicationsAdv. Math.201222112510.1016/j.aim.2011.08.009 DerksenHarmEggermontRob HSnowdenAndrewTopological noetherianity for cubic polynomialsAlgebra Number Theory201711921972212373546710.2140/ant.2017.11.2197 RondGuillaumeArtin approximationJ. Singul.201817108192381313310.5427/jsing.2018.17g BrouwerAndries EDraismaJanEquivariant Gröbner bases and the two-factor modelMath. Comput.2011801123113310.1090/S0025-5718-2010-02415-9 RaicuClaudiu3×3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$3 \times 3$$\end{document} minors of catalecticantsMath. Res. Lett.2013204745756318803010.4310/MRL.2013.v20.n4.a10 9619_CR14 9619_CR15 Claudio Procesi (9619_CR21) 2007 Andries E Brouwer (9619_CR4) 2011; 80 9619_CR16 9619_CR12 9619_CR13 9619_CR1 9619_CR2 Jan Draisma (9619_CR9) 2019; 32 9619_CR3 9619_CR20 9619_CR5 Eric M Friedlander (9619_CR11) 1997; 127 9619_CR6 9619_CR7 Daniel Erman (9619_CR10) 2019; 218 Guillaume Rond (9619_CR23) 2018; 17 Christopher J Hillar (9619_CR17) 2012; 221 9619_CR18 Harm Derksen (9619_CR8) 2017; 11 9619_CR19 Claudiu Raicu (9619_CR22) 2013; 20 |
| References_xml | – reference: M. J. Greenberg. Rational points in Henselian discrete valuation rings. Publ. Math., Inst. Hautes Étud. Sci., 31:563–568, 1966. https://doi.org/10.1007/BF02684802. – reference: RaicuClaudiu3×3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$3 \times 3$$\end{document} minors of catalecticantsMath. Res. Lett.2013204745756318803010.4310/MRL.2013.v20.n4.a10 – reference: Arthur Bik, Jan Draisma, Rob H. Eggermont, and Andrew Snowden. The geometry of polynomial representations. Int. Math. Res. Not., 2021. To appear, arXiv:2105.12621. – reference: Gert-Martin Greuel and Gerhard Pfister. A Singular introduction to commutative algebra. Springer, Berlin, extended edition, 2008. With contributions by Olaf Bachmann, Christoph Lossen and Hans Schönemann. – reference: Toby Ord. How to simulate everything (all at once). URL: http://www.amirrorclear.net/academic/ideas/simulation/index.html. – reference: William Fulton and Joe Harris. Representation Theory. A First Course. Number 129 in Graduate Texts in Mathematics. Springer-Verlag, New York, 1991. – reference: RondGuillaumeArtin approximationJ. Singul.201817108192381313310.5427/jsing.2018.17g – reference: David A. Cox, John Little, and Donal O’Shea. Using algebraic geometry, volume 185 of Graduate Texts in Mathematics. Springer, New York, second edition, 2005. – reference: David A. Cox, John Little, and Donal O’Shea. Ideals, varieties, and algorithms. Undergraduate Texts in Mathematics. Springer, Cham, fourth edition, 2015. An introduction to computational algebraic geometry and commutative algebra. https://doi.org/10.1007/978-3-319-16721-3. – reference: DraismaJanTopological Noetherianity of polynomial functorsJ. Am. Math. Soc.201932691707398198610.1090/jams/923 – reference: J.M. Landsberg and Giorgio Ottaviani. Equations for secant varieties of Veronese and other varieties. Ann. Mat. Pura Appl. (4), 192(4):569–606, 2013. – reference: Arthur Bik. Strength and noetherianity for infinite tensors. PhD thesis, Universität Bern, 2020. – reference: Christopher J. Hillar, Robert Krone, and Anton Leykin. EquivariantGB Macaulay2 package. Webpage: http://people.math.gatech.edu/~rkrone3/EquivariantGB.html, 2013. – reference: Daniel E. Cohen. Closure relations, Buchberger’s algorithm, and polynomials in infinitely many variables. In Computation theory and logic, volume 270 of Lect. Notes Comput. Sci., pages 78–87, 1987. – reference: FriedlanderEric MSuslinAndreiCohomology of finite group schemes over a fieldInvent. Math.19971272209270142761810.1007/s002220050119 – reference: ErmanDanielSamSteven VSnowdenAndrewBig polynomial rings and Stillman’s conjectureInvent. Math.20192182413439401170310.1007/s00222-019-00889-y – reference: DerksenHarmEggermontRob HSnowdenAndrewTopological noetherianity for cubic polynomialsAlgebra Number Theory201711921972212373546710.2140/ant.2017.11.2197 – reference: ProcesiClaudioLie Groups. An Approach through Invariants and Representations2007New YorkSpringer – reference: Arthur Bik, Jan Draisma, Rob H. Eggermont, and Andrew Snowden. Uniformity for limits of tensors. 2023. Preprint, arXiv:2305.19866. – reference: HillarChristopher JSullivantSethFinite Gröbner bases in infinite dimensional polynomial rings and applicationsAdv. Math.201222112510.1016/j.aim.2011.08.009 – reference: Christopher J. Hillar, Robert Krone, and Anton Leykin. Equivariant Gröbner bases. In The 50th anniversary of Gröbner bases. Proceedings of the 8th Mathematical Society of Japan-Seasonal Institute (MSJ-SI 2015), Osaka, Japan, July 1–10, 2015, pages 129–154. Tokyo: Mathematical Society of Japan (MSJ), 2018. – reference: BrouwerAndries EDraismaJanEquivariant Gröbner bases and the two-factor modelMath. Comput.2011801123113310.1090/S0025-5718-2010-02415-9 – reference: Teresa Krick and Alessandro Logar. An algorithm for the computation of the radical of an ideal in the ring of polynomials. In Applied algebra, algebraic algorithms and error-correcting codes (New Orleans, LA, 1991), volume 539 of Lecture Notes in Comput. Sci., pages 195–205. Springer, Berlin, 1991. https://doi.org/10.1007/3-540-54522-0_108. – ident: 9619_CR16 – volume: 221 start-page: 1 year: 2012 ident: 9619_CR17 publication-title: Adv. Math. doi: 10.1016/j.aim.2011.08.009 – volume: 11 start-page: 2197 issue: 9 year: 2017 ident: 9619_CR8 publication-title: Algebra Number Theory doi: 10.2140/ant.2017.11.2197 – volume: 32 start-page: 691 year: 2019 ident: 9619_CR9 publication-title: J. Am. Math. Soc. doi: 10.1090/jams/923 – volume: 127 start-page: 209 issue: 2 year: 1997 ident: 9619_CR11 publication-title: Invent. Math. doi: 10.1007/s002220050119 – ident: 9619_CR18 doi: 10.1007/3-540-54522-0_108. – ident: 9619_CR1 – ident: 9619_CR2 – ident: 9619_CR14 – ident: 9619_CR15 – ident: 9619_CR12 – ident: 9619_CR6 – volume: 218 start-page: 413 issue: 2 year: 2019 ident: 9619_CR10 publication-title: Invent. Math. doi: 10.1007/s00222-019-00889-y – ident: 9619_CR3 – ident: 9619_CR7 doi: 10.1007/978-3-319-16721-3. – volume: 80 start-page: 1123 year: 2011 ident: 9619_CR4 publication-title: Math. Comput. doi: 10.1090/S0025-5718-2010-02415-9 – volume: 20 start-page: 745 issue: 4 year: 2013 ident: 9619_CR22 publication-title: Math. Res. Lett. doi: 10.4310/MRL.2013.v20.n4.a10 – ident: 9619_CR19 doi: 10.1007/s10231-011-0238-6 – volume-title: Lie Groups. An Approach through Invariants and Representations year: 2007 ident: 9619_CR21 – volume: 17 start-page: 108 year: 2018 ident: 9619_CR23 publication-title: J. Singul. doi: 10.5427/jsing.2018.17g – ident: 9619_CR5 doi: 10.1007/3-540-18170-9_156 – ident: 9619_CR13 doi: 10.1007/BF02684802. – ident: 9619_CR20 |
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| SubjectTerms | Algorithms Applications of Mathematics Computer Science Economics Linear and Multilinear Algebras Math Applications in Computer Science Mathematics Mathematics and Statistics Matrix Theory Numerical Analysis Parameterization Polynomials |
| Title | Implicitisation and Parameterisation in Polynomial Functors |
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