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Higher order polar germs. (English) Zbl 0985.14012

This article is about certain properties of higher order polars of a germ of complex analytic curve $C$, say defined near the origin of ${ℂ}^{2}$ by a convergent power series $f\left(x,y\right)$. Recall that the first $x$-polar of $f$ is the germ ${P}_{x}\left(f\right)$ defined by the analytic equation $\partial f/\partial y=0$. If the $y$-axis $Y$ is not an irreducible component of $C$ then $\partial f/\partial y$ is not identically zero, so ${P}_{x}\left(f\right)$ is defined. Then it is not hard to see that $Y$ is not an irreducible component of the first polar and the process may be iterated, getting the second polar ${P}_{x}^{\left(2\right)}\left(f\right)$. Repeating we may define $x$-polars ${P}_{x}^{\left(r\right)}\left(f\right)$ of any order $r$. The main result obtained in this paper is of the following type. Let $C$ be an irreducible germ of plane complex analytic curve (a branch) defined by $f=0$, having a Puiseux series $s=\sum _{i\ge 1}{a}_{i}{x}^{i/n}$, with characteristic exponents ${m}_{1}/n,\cdots ,{m}_{k}/n$, so that if ${n}_{i}:=\text{GCD}\left\{n,{m}_{1},\cdots ,{m}_{i}\right\}$ then ${n}_{k}=1$. If $1\le r, let $u\left(r\right)$ be an integer such that ${n}_{u\left(r\right)-1}>r\ge {n}_{u\left(r\right)}$. Then the germ ${P}_{x}^{\left(r\right)}\left(f\right)$ admits a decomposition ${P}_{x}^{\left(r\right)}\left(f\right)={D}_{1}^{\left(r\right)}+\cdots +{D}_{u\left(r\right)}^{\left(r\right)}$, where the intersection multiplicities $\left[{D}_{i}^{\left(r\right)}·Y\right]$, $1\le i can be expressed in terms of $r,n,{n}_{1},\cdots ,{n}_{u\left(r\right)-1}$ (explicit formulas are given). Moreover, for all $i$, each branch of ${D}_{i}^{\left(r\right)}$ has a Puiseux series which coincides with $s$ up to terms of degree ${m}_{i}-1/n$ (but not for higher degree terms).

The author shows some interesting consequences of this rather technical result. For instance:

(a) an explicit formula for the intersection number $\left[C·B\right]$, where $B$ is any branch of ${P}_{x}^{\left(r\right)}{}_{i}\left(f\right)$, in terms of $\left[B,Y\right]$, $n,{n}_{1},\cdots ,{n}_{i-1},{m}_{1},\cdots ,{m}_{i}$;

(b) with notation as above, the numbers $\left[C·B\right]/\left[Y·B\right]$ coincide with the first $u\left(r\right)$ polar invariants of $C$ (introduced by Teissier);

(c) a new proof of a formula expressing $\left[C·{P}_{x}^{\left(r\right)}\left(f\right)\right]$ in terms of $n$, the ${n}_{i}$’s and the ${m}_{j}$’s, obtained before by A. Dickenstein and C. Sessa [Manuscr. Math. 37, 1-9, (1982; Zbl 0496.14018)]. A special case of (a) gives us a new proof of a result of M. Merle [Invent. Math. 41, 103-111 (1997; Zbl 0371.14003)], expressing the polar invariants of $C$ in terms of its characteristic exponents. By means of examples the author shows that similar results for reducible germs $C$ are no longer valid.

##### MSC:
 14H20 Singularities, local rings 32B10 Germs of analytic sets, local parametrization 32S05 Local singularities (analytic spaces) 14B05 Singularities (algebraic geometry) 32S10 Invariants of analytic local rings