Polyhedral geometry in Groebner theory

Winter Semester 2019/20

Introduction

Solving systems of polynomial equations is one of the most fundamental and important problems in mathematics. Given a system of polynomial equations, the following problems are studied:
(1). How to find exact solutions of this system?
(2). Given another polynomial, does it vanish on the solutions of the system?
These problems can be translated into the language of ideals in a polynomial ring.

Gröbner bases, which are bases of these ideals, are introduced by Hironaka (1964) and Buchberger (1965), as a mixture of the Euclidean division of polynomials, the Gauß elimination of linear equations and the Dantzig simplex algorithm in linear programming. Gröbner bases give a „computational“ answer to these problems.

The theory of Gröbner bases has various applications in Algebraic Geometry, Computational Algebra, Representation Theory, Integral Programming, etc...

In the first part of the lecture we will introduce the Gröbner basis of an ideal in a polynomial ring. With the help of Gröbner theory, we then associate various polyhedral objects (cones, polytopes, etc...) to ideals, and explain why this is helpful in the study of systems of polynomial equations. If time permits, we plan to discuss some recent developments such as Khovanskii bases and tropical dualities.

ILIAS

The lecture (Vorlesung) and the exercise (Übung) will be on ILIAS:

ILIAS link to the Vorlesung: click here

ILIAS link to the Übung: click here

Informations

  • The exam will be on 05.02.2019 from 14 to 16.30, place to be announced.
  • The exercise (Übung) will be held on Monday 16-17.30 in Übungsraum 2 (Gyrhofstr. 8a/b, EG).
  • The lectures will be held on Wednesday 10h-11h30 and Thursday 14h-15h30 in Cohen-Vossen Raum (3.13). Office hour will be on Wednesday 14h-15h or by appointment.

    Contents

  • Lecture 1 (16.10.2019): Introduction.
  • Lecture 2 (17.10.2019): Linear ideals.
  • Lecture 3 (23.10.2019): Monomial orderings, valuation.
  • Lecture 4 (24.10.2019): Monomial ideals, division algorithm.
  • Lecture 5 (30.10.2019): Division algorithm, Groebner basis.
  • Lecture 6 (31.10.2019): Groebner basis: existence.
  • Lecture 7 (06.11.2019): Buchberger criterion.
  • Lecture 8 (07.11.2019): Buchberger algorithm, minimal Groebner basis.
  • Lecture 9 (13.11.2019): Groebner basis: unicity.
  • Lecture 10 (14.11.2019): Examples and applications (I).
  • Lecture 11 (20.11.2019): Examples and applications (II).
  • Lecture 12 (21.11.2019): Examples and applications (III).
  • Lecture 13 (27.11.2019): Standard monomial bases, universality.
  • Lecture 14 (28.11.2019): Universal Groebner bases.
  • Lecture 15 (04.12.2019): Linear ideals and Plücker ideals.
  • Lecture 16 (05.12.2019): Polyhedral cones, Weyl-Minkowski duality, Farkas Lemma.
  • Lecture 17 (11.12.2019): Representation theorem of initial ideals.
  • Lecture 18 (12.12.2019): Groebner fan: affine case (I).
  • Lecture 19 (18.12.2019): Polyhedra and polytopes, face structure.
  • Lecture 20 (19.12.2019): Newton polytopes, examples on tropical duality.
  • Lecture 21 (08.01.2020): More on polyhedral geometry of complexes.
  • Lecture 22 (09.01.2020): Groebner fan: affine case (II).
  • Lecture 23 (15.01.2020): Wall crossing in Groebner fan.
  • Lecture 24 (16.01.2020): Groebner fan: projective case.
  • Lecture 25 (22.01.2020): State polytope, determinantal ideal.
  • Lecture 26 (23.01.2020): Vector partition, fibre walk, weak Bruhat walk, Groebner fan in integer programming, Conti-Traverso.
  • Lecture 27 (29.01.2020): Revision.
  • Lecture 28 (30.01.2020): Office hour session.

    References

    [1]. Adams, William W.; Loustaunau, Philippe. An introduction to Gröbner bases. Graduate Studies in Mathematics, 3. American Mathematical Society, Providence, RI, 1994. xiv+289 pp. ISBN: 0-8218-3804-0.

    [2]. Cox, David A.; Little, John; O'Shea, Donal. Ideals, varieties, and algorithms. An introduction to computational algebraic geometry and commutative algebra. Fourth edition. Undergraduate Texts in Mathematics. Springer, Cham, 2015. xvi+646 pp. ISBN: 978-3-319-16720-6; 978-3-319-16721-3.

    [3]. Kaveh, Kiumars.; Manon, Christopher. Khovanskii bases, higher rank valuations and tropical geometry, to appear in SIAM Journal on Applied Algebra and Geometry (SIAGA), 2019. https://arxiv.org/pdf/1610.00298.pdf

    [4]. Ohsugi, Hidefumi. Convex polytopes and Gröbner bases. Gröbner bases, 223--278, Springer, Tokyo, 2013.

    [5]. Sturmfels, Bernd. Gröbner bases and convex polytopes. University Lecture Series, 8. American Mathematical Society, Providence, RI, 1996. xii+162 pp.

    • Arbeitsgruppe Algebra und Zahlentheorie, Mathematisches Institut, Universität zu Köln