Upper bounding the coefficients of Tutte polynomials with a variable fixed

The Tutte polynomial is a fundamental invariant of graphs and matroids. For a matroid M of rank r, the Tutte polynomial T_M(x,y) of M serves as a generalization of the characteristic polynomial chi_M(x), with the relationship given by chi_M(x) = (-1)^r T_M(1-x,0). For any real number t satisfying 0 <= t <= 1, under suitable exclusion hypotheses on M, we establish an upper bound on the coefficients of the polynomial T_M(x,t) (resp., T_M(t,y)) in terms of girth g and the number of circuits of size g (resp., maximum cardinality h of hyperplanes and the number of hyperplanes of size h). The resulting characteristic polynomial bound extends the bound of Delaplane and Wu [Discrete Math. 312(12-13) (2012) 2062-2068] to simple Ug-1,g+1-free matroids satisfying n >= r + c_g, thereby giving a partial affirmative answer to the question of extending their upper bound beyond binary matroids.