An Inequality for Cosine Polynomials

An inequality for chromatic polynomials

Woodall, D.R., An inequality for chromatic polynomials, Discrete Mathematics 101 (1992) 327-331. It is proved that if P(G, t) is the chromatic polynomial of a simple graph G with II vertices, m edges, c components and b blocks, and if t S 1, then IP(G, t)/ 2 1t’(t l)hl(l + ys + ys2+ . + yF’ +spl), where y = m n + c, p = n c b and s = 1 t. Equality holds for several classes of graphs with few ci...

An inequality for Tutte polynomials

Let G be a graph without loops or bridges and a, b be positive real numbers with b ≥ a(a + 2). We show that the Tutte polynomial of G satisfies the inequality TG(b, 0)TG(0, b) ≥ TG(a, a). Our result was inspired by a conjecture of Merino and Welsh that TG(1, 1) ≤ max{TG(2, 0), TG(0, 2)}.

An Effective Lojasiewicz Inequality for Real Polynomials

Example 1. Set f1 = x d 1 and fi = xi−1 − x d i for i = 2, . . . , n. Then Φ(x) := maxi{|fi(x)|} > 0 for x 6= 0. Let p(t) = (t d , t n−2 , . . . , t). Then limt→0 ||p(t)||/|t| = 1 and Φ(p(t)) = t d . Thus the Lojasiewicz exponent is ≥ d. (In fact it equals d.) This works both over R and C. In the real case set F = ∑ f 2 i . Then degF = 2d, F has an isolated real zero at the origin and the Lojas...

An Inequality About Factors of Multivariate Polynomials

We give a bound on the Euclidean norm of factors of multivariate polynomials. The result is a simple extension of the bivariate case given by Coron, which is an extension of the univariate case given by Mignotte. We use the result to correct a proof by Ernst et al., regarding computing small integer solutions of certain trivariate polynomials.

Turan’s Inequality for Appell Polynomials