5.1 A.The polynomial function f(x)has a zero at x=2 with multiplicity 3.We know that (a) since 3 is an odd number, the graph touches but does not cross the x-axis.(b) since 3 is an odd number, the graph crosses
and prove that it is a rational function (oft). We also show that for large enoughn, the functionis a polynomial, whose leading termis independent of the choice of the witnessV(Theorem6.6): thus,is an invariant ofMthat measures the asymptotic growth ofonM, and refines both the algebraic ...
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Generating functionFinitely generated moduleHilbert seriesGraded polynomial ringIn this note, we give examples of formal power series satisfying certain conditions that cannot be realized as Hilbert series of finitely generated modules. This answers to the negative a question raised in a recent article ...
For any integer g≥3 there exists a graph G with girth g and an N∈N such that PDP(G,m)<P(G,m) whenever m≥N. Since there are examples of large families of graphs that have a DP color function that is eventually strictly smaller than the corresponding chromatic polynomial, the follo...
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