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Abstract It is well known that for two univariate polynomials over the complex number field the number of their common roots is equal to the order of their resultant. In this paper, we show that this fundamental relationship still holds for the tropical polynomials under suitable adaptation of the notion of order, if the roots are simple and non-zero. [ABSTRACT FROM AUTHOR]

Recently, M. Abért and T. Hubai studied the following problem. The chromatic measure of a finite simple graph is defined to be the uniform distribution on its chromatic roots. Abért and Hubai proved that for a Benjamini–Schramm convergent sequence of finite graphs, the chromatic measures converge in holomorphic moments. They also showed that the normalized logarithm of the chromatic polynomial converges to a harmonic real function outside a bounded disc. In this paper we generalize their work to a wide class of graph polynomials, namely, multiplicative graph polynomials of bounded exponential type. A special case of our results is that for any fixed complex number v 0 the measures arising from the Tutte polynomial Z G n ( z , v 0 ) converge in holomorphic moments if the sequence ( G n ) of finite graphs is Benjamini–Schramm convergent. This answers a question of Abért and Hubai in the affirmative. Even in the original case of the chromatic polynomial, our proof is considerably simpler. [ABSTRACT FROM AUTHOR]

A polynomial is called unimodular if each of its coefficients is a complex number of modulus 1. A polynomial P of the form P ( z ) = ∑ j = 0 n a j z j is called conjugate reciprocal if a n − j = a ‾ j , a j ∈ C for each j = 0 , 1 , … , n . Let ∂ D be the unit circle of the complex plane. We prove that there is an absolute constant ε > 0 such that max z ∈ ∂ D ⁡ | f ( z ) | ≥ ( 1 + ε ) 4 / 3 m 1 / 2 , for every conjugate reciprocal unimodular polynomial of degree m . We also prove that there is an absolute constant ε > 0 such that M q ( f ′ ) ≤ exp ⁡ ( ε ( q − 2 ) / q ) 1 / 3 m 3 / 2 , 1 ≤ q < 2 , and M q ( f ′ ) ≥ exp ⁡ ( ε ( q − 2 ) / q ) 1 / 3 m 3 / 2 , 2 < q , for every conjugate reciprocal unimodular polynomial of degree m , where M q ( g ) = ( 1 2 π ∫ 0 2 π | g ( e i t ) | q d t ) 1 / q , q > 0 . [ABSTRACT FROM AUTHOR]

Using the basic fact that any formal power series over the real or complex number field can always be expressed in terms of given polynomials {p n (t)}, where p n (t) is of degree n, we extend the ordinary Riordan array (resp. Riordan group) to a generalized Riordan array (resp. generalized Riordan group) associated with {p n (t)}. As new application of the latter, a rather general Vandermonde-type convolution formula and certain of its particular forms are presented. The construction of the Abel type identities using the generalized Riordan arrays is also discussed. [ABSTRACT FROM AUTHOR]