Polynomials

Irreducibility Criteria for Sums of Two Relatively Prime Polynomials

“…We will first recall some facts about Newton polygons (see for instance [1]), that will be required in the proof of Lemma 1.4. So let p be a fixed prime number, and let f (X) = n i=0 a i X i be a polynomial with integer coefficients, a 0 a n = 0.…”

Parallel machine scheduling with high multiplicity

“…On each machine, the jobs are processed in SPT (shortest processing time) order. Let p i[ j] = processing time of the j th shortest job type on machine i, i.e., p i [1] ≤ p i [2] ≤ · · · ≤ p i[g] , x i j = number of jobs of type j to be processed on machine i, x i[ j] = number of jobs of j th shortest job type to be processed on machine i, x i0 = number of dummy jobs to be processed on machine i, and n 0 = (m − 1)n.…”

“…Since S is a matrix of supply constraints, there is exactly one 1 in each column. For each row, d i of D, there are g + 1 consecutive ones corresponding to variables x i[0] , x i [1] , . .…”

On the Roots of Chromatic Polynomials

“…A number of papers have considered the location of the roots of the chromatic polynomial of a graph. Birkhoff and Lewis [7] showed that the chromatic polynomial of any plane triangulation has no roots in the intervals (& , 0), (0, 1), (1,2), and [5, ), and Woodall [22] improved this by showing that in fact there are no roots in (2, 2.546602...) (the latter being the smallest nonintegral real root of the chromatic polynomial of the octahedron). It is well known (see [20]) that no graph has a root of its chromatic polynomial in (& , 0) or (0, 1).…”