On the Łojasiewicz exponent of the gradient of a polynomial function

• Autorzy: Andrzej Lenarcik
• Języki publikacji: EN

Abstrakty

EN

Let $h = ∑ h_{αβ} X^αY^β$ be a polynomial with complex coefficients. The Łojasiewicz exponent of the gradient of h at infinity is the least upper bound of the set of all real λ such that $|grad h(x,y)| ≥ c|(x,y)|^λ$ in a neighbourhood of infinity in ℂ², for c > 0. We estimate this quantity in terms of the Newton diagram of h. Equality is obtained in the nondegenerate case.

Słowa kluczowe

EN

polynomial mapping; Łojasiewicz exponent; Newton diagram

Kategorie tematyczne

• 14E05: Rational and birational maps

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Annales Polonici Mathematici
• Rocznik: 1999
• Tom: 71
• Numer: 3
• Strony: 211-239

Daty

wydano

1999

otrzymano

1997-06-17

poprawiono

1998-06-08

poprawiono

1999-04-06

Twórcy

autor

Andrzej Lenarcik

• Kielce University of Technology, Aleja Tysiąclecia Państwa Polskiego 7, 25-314 Kielce, Poland

Bibliografia

• [A] M. Artin, On the solution of analytic equations, Invent. Math. 5 (1968), 277-291.
• [BK] E. Brieskorn und H. Knörrer, Ebene algebraische Kurven, Birkhäuser, 1981.
• [CN-H] P. Cassou-Noguès et Há Huy Vui, Sur le nombre de Łojasiewicz à l'infini d'un polynôme, Ann. Polon. Math. 62 (1995), 23-44.
• [ChK1] J. Chądzyński and T. Krasiński, Exponent of growth of polynomial mappings of ℂ² into ℂ², in: Singularities, Banach Center Publ. 20, PWN, Warszawa, 1988, 147-160.
• [ChK2] J. Chądzyński and T. Krasiński, On the Łojasiewicz exponent at infinity for polynomial mappings of ℂ² into ℂ² and components of polynomial automorphisms of ℂ², Ann. Polon. Math. 57 (1992), 291-302.
• [L] A. Lenarcik, On the Łojasiewicz exponent of the gradient of a holomorphic function, in: Singularities Symposium - Łojasiewicz 70, Banach Center Publ. 44, Inst. Math., Polish Acad. Sci., Warszawa, 1998, 149-166.
• [Pł1] A. Płoski, On the growth of proper polynomial mappings, Ann. Polon. Math. 45 (1985), 297-309.
• [Pł2] A. Płoski, Newton polygons and the Łojasiewicz exponent of a holomorphic mapping of ℂ², Ann. Polon. Math. 51 (1990), 275-281.
• [W] R. Walker, Algebraic Curves, Princeton Univ. Press, 1950.