Minimal orderings and quadratic forms on a free module over a supertropical semiring

Zur Izhakian, Manfred Knebusch, Louis Rowen

Research output: Contribution to journalArticle

3 Citations (Scopus)
4 Downloads (Pure)

Abstract

This paper is a sequel to [6], in which we introduced quadratic forms on a module over a supertropical semiring R and analyzed the set of bilinear companions of a single quadratic form V→R in case the module V is free. Any (semi)module over a semiring gives rise to what we call its minimal ordering, which is a partial order iff the semiring is “upper bound.” Any polynomial map q (or quadratic form) then induces a pre-order, which can be studied in terms of “q-minimal elements,” which are elements a which cannot be written in the form b+c where b<a but q(b)=q(a). We determine the q-minimal elements by examining their support.
But the class of all polynomial maps (in up to rank(V) variables) is itself a module over R, so the basic properties of the minimal ordering are applied to this R -module, or its submodule Quad(V) consisting of quadratic forms on V. This is a significant initial step in the classification of quadratic forms over semirings arising in tropical mathematics.
Quad(V) is the sum of two disjoint submodules QL(V) and Rig(V), consisting of the quasilinear and the rigid quadratic forms on V respectively (cf. [6]). Both QL(V) and Rig(V) are free with explicitly known bases, but Quad(V) itself is almost never free.
Original languageEnglish
Pages (from-to)420-461
Number of pages42
Journal Linear Algebra and its Applications
Volume507
Early online date15 Jun 2016
DOIs
Publication statusPublished - 15 Oct 2016

Fingerprint

Semiring
Quadratic form
Polynomials
Module
Polynomial Maps
Preorder
Partial Order
Disjoint
Upper bound

Keywords

  • Tropical algebra
  • Supertropical modules
  • Bilinear forms
  • Quadratic forms
  • Quadratic pairs
  • Supertropicalization
  • Varieties

Cite this

Minimal orderings and quadratic forms on a free module over a supertropical semiring. / Izhakian, Zur; Knebusch, Manfred; Rowen, Louis.

In: Linear Algebra and its Applications, Vol. 507, 15.10.2016, p. 420-461.

Research output: Contribution to journalArticle

@article{47031ead662c45cc9305dff35374e4b6,
title = "Minimal orderings and quadratic forms on a free module over a supertropical semiring",
abstract = "This paper is a sequel to [6], in which we introduced quadratic forms on a module over a supertropical semiring R and analyzed the set of bilinear companions of a single quadratic form V→R in case the module V is free. Any (semi)module over a semiring gives rise to what we call its minimal ordering, which is a partial order iff the semiring is “upper bound.” Any polynomial map q (or quadratic form) then induces a pre-order, which can be studied in terms of “q-minimal elements,” which are elements a which cannot be written in the form b+c where bBut the class of all polynomial maps (in up to rank(V) variables) is itself a module over R, so the basic properties of the minimal ordering are applied to this R -module, or its submodule Quad(V) consisting of quadratic forms on V. This is a significant initial step in the classification of quadratic forms over semirings arising in tropical mathematics.Quad(V) is the sum of two disjoint submodules QL(V) and Rig(V), consisting of the quasilinear and the rigid quadratic forms on V respectively (cf. [6]). Both QL(V) and Rig(V) are free with explicitly known bases, but Quad(V) itself is almost never free.",
keywords = "Tropical algebra, Supertropical modules, Bilinear forms, Quadratic forms, Quadratic pairs, Supertropicalization, Varieties",
author = "Zur Izhakian and Manfred Knebusch and Louis Rowen",
note = "The authors thank the referee for detailed suggestions to improve the exposition.",
year = "2016",
month = "10",
day = "15",
doi = "10.1016/j.laa.2016.06.001",
language = "English",
volume = "507",
pages = "420--461",
journal = "Linear Algebra and its Applications",
issn = "0024-3795",
publisher = "Elsevier",

}

TY - JOUR

T1 - Minimal orderings and quadratic forms on a free module over a supertropical semiring

AU - Izhakian, Zur

AU - Knebusch, Manfred

AU - Rowen, Louis

N1 - The authors thank the referee for detailed suggestions to improve the exposition.

PY - 2016/10/15

Y1 - 2016/10/15

N2 - This paper is a sequel to [6], in which we introduced quadratic forms on a module over a supertropical semiring R and analyzed the set of bilinear companions of a single quadratic form V→R in case the module V is free. Any (semi)module over a semiring gives rise to what we call its minimal ordering, which is a partial order iff the semiring is “upper bound.” Any polynomial map q (or quadratic form) then induces a pre-order, which can be studied in terms of “q-minimal elements,” which are elements a which cannot be written in the form b+c where bBut the class of all polynomial maps (in up to rank(V) variables) is itself a module over R, so the basic properties of the minimal ordering are applied to this R -module, or its submodule Quad(V) consisting of quadratic forms on V. This is a significant initial step in the classification of quadratic forms over semirings arising in tropical mathematics.Quad(V) is the sum of two disjoint submodules QL(V) and Rig(V), consisting of the quasilinear and the rigid quadratic forms on V respectively (cf. [6]). Both QL(V) and Rig(V) are free with explicitly known bases, but Quad(V) itself is almost never free.

AB - This paper is a sequel to [6], in which we introduced quadratic forms on a module over a supertropical semiring R and analyzed the set of bilinear companions of a single quadratic form V→R in case the module V is free. Any (semi)module over a semiring gives rise to what we call its minimal ordering, which is a partial order iff the semiring is “upper bound.” Any polynomial map q (or quadratic form) then induces a pre-order, which can be studied in terms of “q-minimal elements,” which are elements a which cannot be written in the form b+c where bBut the class of all polynomial maps (in up to rank(V) variables) is itself a module over R, so the basic properties of the minimal ordering are applied to this R -module, or its submodule Quad(V) consisting of quadratic forms on V. This is a significant initial step in the classification of quadratic forms over semirings arising in tropical mathematics.Quad(V) is the sum of two disjoint submodules QL(V) and Rig(V), consisting of the quasilinear and the rigid quadratic forms on V respectively (cf. [6]). Both QL(V) and Rig(V) are free with explicitly known bases, but Quad(V) itself is almost never free.

KW - Tropical algebra

KW - Supertropical modules

KW - Bilinear forms

KW - Quadratic forms

KW - Quadratic pairs

KW - Supertropicalization

KW - Varieties

U2 - 10.1016/j.laa.2016.06.001

DO - 10.1016/j.laa.2016.06.001

M3 - Article

VL - 507

SP - 420

EP - 461

JO - Linear Algebra and its Applications

JF - Linear Algebra and its Applications

SN - 0024-3795

ER -