Weak Tutte Functions of Matroids

We find the universal module w(M) for functions (that need not be invariants) of finite matroids, defined on a minor-closed class M and with values in any module L over any commutative and unitary ring, that satisfy a parametrized deletion-contraction identity, F (M) = δeF (M r e) + γeF (M/e), when e is neither a loop nor a coloop. (F is called a (parametrized) weak Tutte function.) Within the universal module each matroid has a linear form t(M), its Tutte form, such that, for every weak Tutte function, F (M) is an evaluation of t(M) through a unique homomorphism w(M)→ L. If L is an algebra and we also require multiplicativity, F (M1 ⊕M2) = F (M1)F (M2), then there is a universal commutative algebra W(M) (previously studied in special cases by Bollobás and Riordan) which is the image of the Tutte module w(M) under a canonical homomorphism. The image of t(M) is the Tutte polynomial T (M), an element of W(M). T (M) may not be a true polynomial because its variables may have relations. Nevertheless, T (M) is universal for multiplicative weak Tutte functions; that is, F (M) is an evaluation of t(M) through a unique homomorphism W(M) → L. If the canonical homomorphism w(M) → W(M) is injective, then T (M) is universal for all weak Tutte functions, not only multiplicative ones, thus resembling the classical Tutte polynomial that is universal for nonmultiplicative as well as multiplicative Tutte invariants of finite matroids. If the homomorphism is not injective, then the Tutte polynomial is not universal for weak Tutte functions. It is possible for the canonical homomorphism to be noninjective, though Bollobás and Riordan’s work implies it is injective in the cases they treated. From W(M) we deduce similar universal algebraic structures for weak Tutte functions with limited multiplicativity requirements, in particular the separator-strong functions of Ellis-Monaghan and Traldi. Version compiled July 31, 2006.