Galois groups of multivariate Tutte polynomials
نویسندگان
چکیده
The multivariate Tutte polynomial ẐM of a matroid M is a generalization of the standard two-variable version, obtained by assigning a separate variable ve to each element e of the ground set E. It encodes the full structure of M . Let v = {ve}e∈E , let K be an arbitrary field, and suppose M is connected. We show that ẐM is irreducible over K(v), and give three self-contained proofs that the Galois group of ẐM over K(v) is the symmetric group of degree n, where n is the rank of M . An immediate consequence of this result is that the Galois group of the multivariate Tutte polynomial of any matroid is a direct product of symmetric groups. Finally, we conjecture a similar result for the standard Tutte polynomial of a connected matroid.
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