zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
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 $\bbfC^2$ by a convergent power series $f(x,y)$. Recall that the first $x$-polar of $f$ is the germ $P_x(f)$ 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 $P_x(f)$ 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 $P^{(2)}_x(f)$. Repeating we may define $x$-polars $P^{(r)}_x(f)$ 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 $s=\displaystyle\sum_{i\ge 1}a_i x^{i/n}$, with characteristic exponents $m_1/n,\dots, m_k/n$, so that if $n_i:=\text{GCD}\{n,m_1, \dots,m_i\}$ then $n_k=1$. If $1\le r<n$, let $u(r)$ be an integer such that $n_{u(r)-1} >r\ge n_{u (r)}$. Then the germ $P_x^{(r)}(f)$ admits a decomposition $P_x^{(r)} (f)= D^{(r)}_1 +\cdots+ D^{(r)}_{u(r)}$, where the intersection multiplicities $[D_i^{(r)}.Y]$, $1\le i<u(r)$ can be expressed in terms of $r,n,n_1, \dots, n_{u(r)-1}$ (explicit formulas are given). Moreover, for all $i$, each branch of $D_i^{(r)}$ 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 $[C.B]$, where $B$ is any branch of $P^{(r)}_x{_i(f)}$, in terms of $[B,Y]$, $n,n_1,\dots,n_{i-1},m_1, \dots, m_i$; (b) with notation as above, the numbers $[C.B]/[Y.B]$ coincide with the first $u(r)$ polar invariants of $C$ (introduced by Teissier); (c) a new proof of a formula expressing $[C.P_x^{(r)} (f)]$ in terms of $n$, the $n_i$’s and the $m_j$’s, obtained before by {\it A. Dickenstein} and {\it 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 {\it 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.

14H20Singularities, local rings
32B10Germs of analytic sets, local parametrization
32S05Local singularities (analytic spaces)
14B05Singularities (algebraic geometry)
32S10Invariants of analytic local rings
Full Text: DOI
[1] Brieskorn, E.; Knörrer, H.: Plane algebraic curves. (1986)
[2] Casas-Alvero, E.: Infinitely near imposed singularities and singularities of polar curves. Math. ann. 287, 429-454 (1990) · Zbl 0675.14009
[3] Casas-Alvero, E.: Singularities of plane curves. London math. Soc. lecture note series 276 (2000) · Zbl 0967.14018
[4] Dickenstein, A.; Sessa, C.: An integral criterion for the equivalence of plane curves. Manuscripta math. 37, 1-9 (1982) · Zbl 0496.14018
[5] Enriques, F.; Chisini, O.: Lezioni sulla teoria geometrica delle equazioni e delle funzioni algebriche. (1915) · Zbl 0009.15904
[6] Merle, M.: Invariants polaires des courbes planes. Invent. math. 41, 103-111 (1997) · Zbl 0371.14003
[7] Pham, F.: Deformations equisingulières des idéaux jacobiens des courbes planes. Lect. notes in math. 209 (1971)
[8] Semple, J. G.; Kneebone, G. T.: Algebraic curves. (1959) · Zbl 0105.34301
[9] Teissier, B.: The hunting of invariants in the geometry of discriminants. (1977) · Zbl 0388.32010
[10] Teissier, B.: Variétés polaires I. Invariants polaires des singularités d’hypersurfaces. Invent. math. 40, 267-292 (1977) · Zbl 0446.32002
[11] Zariski, O.: Studies in equisingularity III. Am. J. Math. 90, 961-1023 (1968) · Zbl 0189.21405