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Quadratic and Symmetric Bilinear Forms on Modules with Unique Base Over a Semiring

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Zur Izhakian, Manfred Knebusch, Louis Rowen

Rings and Algebras · Submitted 3 Sep 2015

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim We study quadratic forms on free modules with unique base, the situation that arises in tropical algebra, and prove the analog of Witt's Cancellation Theorem.

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arXiv ID1509.01039 (version 1)
Pith ID2JBQUHL4, Pith's permanent record of this paper and its review
CitationLouis Rowen, Manfred Knebusch, Zur Izhakian. "Quadratic and Symmetric Bilinear Forms on Modules with Unique Base Over a Semiring." arXiv:1509.01039 (2015). https://pith.science/paper/2JBQUHL4
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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 2,350 input tokens.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.77 · human participants 0.01 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.33 · compares with prior results 0.25 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; .

We study quadratic forms on free modules with unique base, the situation that arises in tropical algebra, and prove the analog of Witt's Cancellation Theorem. Also, the tensor product of an indecomposable bilinear module $(U, \gamma)$ with an indecomposable quadratic module $(V,q) $ is indecomposable, with the exception of one case, where two indecomposable components arise.

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#sentencerolethe authors' own workstates the main conclusionstrength
s0 We study quadratic forms on free modules with unique base, the situation that arises in tropical algebra, and prove the analog of Witt's Cancellation Theorem. contribution 0.90 0.94 0.67 asserted 1.00
s1 Also, the tensor product of an indecomposable bilinear module $(U, \gamma)$ with an indecomposable quadratic module $(V,q) $ is indecomposable, with the exception of one case, where two indecomposable components arise. result 1.00 0.90 0.18 asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

The paper
Original abstract

We study quadratic forms on free modules with unique base, the situation that arises in tropical algebra, and prove the analog of Witt's Cancellation Theorem. Also, the tensor product of an indecomposable bilinear module $(U, \gamma)$ with an indecomposable quadratic module $(V,q) $ is indecomposable, with the exception of one case, where two indecomposable components arise.

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1 arXiv paper cite this in 4 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 3 sentences without a confident label

Cites it as support 1 paper
  • By [ref] , the partial pre-order (i) of Example [ref] is a PO iff the semigroup $𝒜$ is ub.

    Algebras with a negation map · Introducing $$-surpassing relations and surpassing PO's · 0.96

What it cites · 19 sentences without a confident label

Cites as support 1 paper
Cites as background 1 paper
  • We recall from [ref] that, when $V$ is free with base $(_i : i ∈I)$, then $q$ admits a (not necessarily unique) balanced companion, i.e., a companion $b : V ×V →R$ such that $b(x,x) = 2q(x)$ for all $x ∈V$, and that it suffices to know for this that $b(_i,_i) = 2q( _i)$ for all $i ∈I$ [ref] .

    Supertropical Quadratic Forms I · Expansions and tensor products · 0.85

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