Polynomial equations with one catalytic variable, algebraic series, and map enumeration
| hal.structure.identifier | Laboratoire Bordelais de Recherche en Informatique [LaBRI] | |
| dc.contributor.author | BOUSQUET-MÉLOU, Mireille | |
| hal.structure.identifier | Théorie des Nombres et Algorithmique Arithmétique [A2X] | |
| dc.contributor.author | JEHANNE, Arnaud | |
| dc.date.issued | 2006 | |
| dc.identifier.issn | 0095-8956 | |
| dc.description.abstractEn | Let $F(t,u)\equiv F(u)$ be a formal power series in $t$ with polynomial coefficients in $u$. Let $F_1 , \ldots, F_k$ be $k$ formal power series in $t$, independent of $u$. Assume all these series are characterized by a polynomial equation $$ P(F(u), F_1, \ldots , F_k, t , u)=0. $$ We prove that, under a mild hypothesis on the form of this equation, these $(k+1)$ series are algebraic, and we give a strategy to compute a polynomial equation for each of them. This strategy generalizes the so-called kernel method, and quadratic method, which apply respectively to equations that are linear and quadratic in $F(u)$. Applications include the solution of numerous map enumeration problems, among which the hard-particle model on general planar maps. | |
| dc.language.iso | en | |
| dc.publisher | Elsevier | |
| dc.subject.en | generating functions | |
| dc.subject.en | enumeration | |
| dc.subject.en | kernel method | |
| dc.subject.en | planar maps | |
| dc.subject.en | functional equations | |
| dc.subject.en | quadratic method | |
| dc.title.en | Polynomial equations with one catalytic variable, algebraic series, and map enumeration | |
| dc.type | Article de revue | |
| dc.subject.hal | Mathématiques [math]/Combinatoire [math.CO] | |
| dc.identifier.arxiv | math.CO/0504018 | |
| bordeaux.journal | Journal of Combinatorial Theory, Series B | |
| bordeaux.page | 623--672. | |
| bordeaux.volume | 96 | |
| bordeaux.peerReviewed | oui | |
| hal.identifier | hal-00004621 | |
| hal.version | 1 | |
| hal.popular | non | |
| hal.audience | Internationale | |
| hal.origin.link | https://hal.archives-ouvertes.fr//hal-00004621v1 | |
| bordeaux.COinS | ctx_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:journal&rft.jtitle=Journal%20of%20Combinatorial%20Theory,%20Series%20B&rft.date=2006&rft.volume=96&rft.spage=623--672.&rft.epage=623--672.&rft.eissn=0095-8956&rft.issn=0095-8956&rft.au=BOUSQUET-M%C3%89LOU,%20Mireille&JEHANNE,%20Arnaud&rft.genre=article |
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