Lift theorems for representations of matroids over pastures

Abstract: Given a partial field P, the Lift Theorem of Pendavingh and van Zwam produces a partial field L(P) and a homomorphism L(P) -> P with the property that if a matroid is representable over P then it is also representable over L(P). We will formulate a generalization of the Pendavingh-van Zwam Lift Theorem to pastures, which generalize both partial fields and hyperfields, and explore some of its combinatorial implications. For certain restricted classes of matroids (e.g. ternary matroids), we obtain a stronger lift theorem which is essentially sharp. Even in the case of partial fields, our method of proof is different from that of Pendavingh and van Zwam and we're able to prove some new results. This is joint work with Oliver Lorscheid.

algebraic geometrycombinatorics

Audience: researchers in the topic

(LAGARTOS) Latin American Real and Tropical Geometry Seminar

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