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**The Möbius function and the characteristic polynomial.**
*(English)*
Zbl 0632.05017

Combinatorial geometries, Encycl. Math. Appl. 29, 114-138 (1987).

[For the entire collection see Zbl 0626.00007.]

The characteristic polynomial p(M;\(\lambda)\) of a matroid M is an analogue of the chromatic polynomial of a graph. While, in general, this polynomial is not associated with colouring anything, it has proved to be a basic matroid invariant. If M is a simplex rank-r matroid representable over GF(q) and \(\psi\) is a representation of M as a restriction of V(r,q), the critical exponent c(M;q) of M is the least number of hyperplanes of V(r,q) whose intersection with \(\psi\) (E(M)) is empty. H. Crapo and G.-C. Rota [On the foundations of combinatorial theory, Stud. Appl. Math. 49, 109-133 (1970; Zbl 0231.05024)] proved that c(M;q) is independent of \(\psi\) by showing that it equals the least positive integer k for which \(p(M;q^ k)\) is positive. This chapter surveys results on the characteristic polynomial, the critical exponent, and two other related invariants: the Möbius invariant, \(\mu\) (M), which equals p(M;0); and Crapo’s beta invariant, \(\beta\) (M), which equals \((-1)^{r(M)-1}\frac{d}{d\lambda}p(M;1)\). Two results that give a flavour for the sort of information provided by these invariants are the following: a binary matroid has critical exponent one if and only if all its circuits are even; and an arbitrary matroid M with two or more elements is connected if and only if \(\beta (M)>0\).

The characteristic polynomial p(M;\(\lambda)\) of a matroid M is an analogue of the chromatic polynomial of a graph. While, in general, this polynomial is not associated with colouring anything, it has proved to be a basic matroid invariant. If M is a simplex rank-r matroid representable over GF(q) and \(\psi\) is a representation of M as a restriction of V(r,q), the critical exponent c(M;q) of M is the least number of hyperplanes of V(r,q) whose intersection with \(\psi\) (E(M)) is empty. H. Crapo and G.-C. Rota [On the foundations of combinatorial theory, Stud. Appl. Math. 49, 109-133 (1970; Zbl 0231.05024)] proved that c(M;q) is independent of \(\psi\) by showing that it equals the least positive integer k for which \(p(M;q^ k)\) is positive. This chapter surveys results on the characteristic polynomial, the critical exponent, and two other related invariants: the Möbius invariant, \(\mu\) (M), which equals p(M;0); and Crapo’s beta invariant, \(\beta\) (M), which equals \((-1)^{r(M)-1}\frac{d}{d\lambda}p(M;1)\). Two results that give a flavour for the sort of information provided by these invariants are the following: a binary matroid has critical exponent one if and only if all its circuits are even; and an arbitrary matroid M with two or more elements is connected if and only if \(\beta (M)>0\).

Reviewer: J.G.Oxley

### MSC:

05B35 | Combinatorial aspects of matroids and geometric lattices |