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Motivic Milnor fiber at infinity and composition with a non-degenerate polynomial. (Fibre de Milnor motivique à l’infini et composition avec un polynôme non dégénéré.) (French. English summary) Zbl 1266.14008
Let $$P$$ be a Laurent polynomial in $$d$$ variables with coefficients in a field of characteristic zero and non-degenerated relative to its Newton polyhedron $$\Gamma$$ at infinity. Let $$f_1,\dots,f_d$$ be non-constant functions defined on smooth varieties. The author computes the motivic Milnor fiber at infinity in the sense [C.R., Math., Acad. Sci. Paris 348, No. 7–8, 419–422 (2010; Zbl 1195.14028)] for the superposition $$P(f_1,\dots, f_d)$$ in terms of generalized motivic nearby cycles at infinity associated with the functions $$f_i$$, polynomial $$P$$ and faces of $$\Gamma$$. In particular, he obtains a formula of Thom-Sebastiani type at infinity for $$P = x_1 + x_2$$, as well as his own formula in the case where $$f_i$$ are coordinates of $$d$$-dimensional torus (Theorem 3.3 in the author’s paper [Bull. Soc. Math. Fr. 140, No. 1, 51–100 (2012; Zbl 1266.14012)]), etc. Then a notion of motivic vanishing cycles of a function $$g$$ for the infinite value is discussed. It turns out that for Laurent polynomial $$g= x_1+\dots+x_n+1/x_1\dots x_n$$ the corresponding spectrum is equal to $$1+t+\dots+t^n,$$ that is, it coincides with the spectrum at infinity of $$g$$ considered by A. Douai and C. Sabbah [Ann. Inst. Fourier 53, No. 4, 1055–1116 (2003; Zbl 1079.32016)].
MSC:
 14D06 Fibrations, degenerations in algebraic geometry 32S55 Milnor fibration; relations with knot theory 14E18 Arcs and motivic integration 14R25 Affine fibrations 32S35 Mixed Hodge theory of singular varieties (complex-analytic aspects)
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References:
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