A c-holomorphic effective Nullstellensatz with parameter

Maciej Denkowski


We prove a local Nullstellensatz with parameter for a continuous family of c-holomorphic functions with an e ective exponent independent of the parameter: the local degree of the cycle of zeroes of the central section. We assume that this central section de nes a proper intersection and we show that we can omit this assumption in case of isolated zeroes.

Słowa kluczowe: Nullstellensatz, c-holomorphic functions

1. Achilles R., Tworzewski P., Winiarski T., On improper isolated intersection in complex analytic geometry, Ann. Polon. Math., LI (1990), 21-36.

2. Chirka E. M., Complex Analytic Sets, Kluwer Acad. Publ., 1989.

3. Cygan E., Nullstellensatz and cycles of zeroes of holomorphic mappings, Ann. Polon. Math., LXXVIII (2002), 181-191.

4. Denkowska Z., Denkowski M. P., Kuratowski convergence and connected components, J. Math. Anal. Appl., 387 no. 1 (2012), 48-65.

5. Denkowski M. P., The  Lojasiewicz exponent of c-holomorphic mappings, Ann. Polon. Math., 87 no. 1 (2005), 63-81.

6. Denkowski M. P., A note on the Nullstellensatz for c-holomorphic functions, Ann. Polon. Math., 90 no. 3 (2007), 219-228.

7. Denkowski M. P., On the complex  Lojasiewicz inequality with parameter, preprint arXiv:1406.1700 (2014).

8. Draper R. N., Intersection theory in analytic geometry, Math. Ann., 180 (1969), 175-204.

9. Galligo A., Gonzalez-Vega L., Lombardi H., Continuity properties for at families of polynomials (I) Continuous parametrizations, J. Pure Appl. Algebra, 184 (2003) 77-103.

10. P loski A., Tworzewski P., E ective Nullstellensatz on analytic and algebraic varieties, Bull. Polish Acad. Sci. Math., 46 (1998), 31-38.

11. Remmert R., Projektionen analytischer Mengen, Math. Ann., 130 (1956), 410-441.

12. Spodzieja S., Multiplicity and the  Lojasiewicz exponent, Ann. Polon. Math., 73 (2000) no. 3, 257-267.

13. Tworzewski P., Intersection theory in complex analytic geometry, Ann. Polon. Math. LXII.2 (1995), 177-191.

14. Tworzewski P., Winiarski T., Continuity of intersection of analytic sets, Ann. Polon. Math., 42 (1983), 387-393.

15. Tworzewski P., Winiarski T., Cycles of zeroes of holomorphic mappings, Bull. Polish Acad. Sci. Math., 37 (1986), 95-101.

16. Whitney H., Complex Analytic Varieties, Addison-Wesley Publ. Co., 1972.

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