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Zbl 0746.14002
Cassou-Noguès, Pierrette
Iterated toric webs and integrals associated to a plane curve. (Entrelacs toriques itérés et intégrales associées à une courbe plane.) (French)
[J] Sémin. Théor. Nombres Bordx., Sér. II 2, No.2, 273-331 (1990). ISSN 0989-5558

This is an expository paper, presenting the background to the author's major results in J. Number Theory 23, 1-54 (1986; Zbl 0584.10022) and Ann. Inst. Fourier 36, No. 4, 1-30 (1986; Zbl 0597.32004). Definitions of knots, links, torus links and iterated torus links, and of their Eisenbud-Neumann diagrams are given and illustrated by examples. Of particular interest to the author are examples obtained by intersection of the zero locus of a polynomial in 2 complex variables with a sphere of very small radius (the local link) or very large radius (the link at infinity). The traditional relations with Newton polygons, and with Puiseux exponents, and the calculation of the Alexander polynomial are described, and extended to the case of the link at infinity. The Seifert form of a link is used to define its multisignature, and this in turn is used in the definition of the ``spectrum''. An extensive set of calculations is presented here: several particular examples, torus knots, the general algebraic knot, the local link for a Newton nondegenerate function, and the global link in the corresponding case. Finally the author considers the integrals $$\align I\sb \infty(f,\beta)(s)&=\int\sp \infty\sb 1\int\sp \infty\sb 1x\sb 1\sp{\beta\sb 1-1}\cdot x\sb 2\sp{\beta\sb 2-1}\cdot f(x\sb 1,x\sb 2)\sp{-s}dx\sb 1dx\sb 2\quad\text{ and } \\ I\sb 0(f,\beta)(s)&=\int\sp 1\sb 0\int\sp 1\sb 0x\sb 1\sp{\beta\sb 1-1}\cdot x\sb 2\sp{\beta\sb 2-1}\cdot f(x\sb 1,x\sb 2)\sp sdx\sb 1dx\sb 2.\endalign$$ These converge in half-planes $\text{Re}(s)>\sigma$ and admit meromorphic extension to $\bbfC$. The poles of $I\sb 0$ where known to be related to the spectrum of the local link of $f$ at (0,0). The new results give a relation between poles of $I\sb \infty$ and the spectrum of the global link. The details are too complicated to summarise here; the main results assume that the coefficients in $f$ are real and positive.
[C.T.C.Wall (Liverpool)]

MSC 2000:: *14C21 Webs, nets
14H20 Singularities, local rings
33C70 Other hypergeometric functions and integrals in several variables

Keywords: local link; link at infinity; Newton polygons; Alexander polynomial; Seifert form

Citations: Zbl 0584.10022; Zbl 0597.32004

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