A Family of Algebraically Closed Fields Containing Polynomials in Several Variables

Abstract

We introduce a family of fields of series with support in strongly convex rational cones. All these fields contain polynomials in several variables. We prove that they are algebraically closed with a construction that is analogous to the Newton polygon for algebraic curves. As a corollary, we show the existence of fractional power solutions with support in cones for systems of equations.

Key Words:

• Newton polygon
• Parametrization
• Puiseux series

2000 Mathematics Subject Classification:

• Primary 32S05, 32B10
• Secondary 14M25

ACKNOWLEDGMENTS

We would like to thank Paloma for playing with Benedetta and Maria while we were working. The first author was partially supported by DGAPA-UNAM: PAPIIT IN105806 and CONACYT 550084.

Notes

Communicated by R. Piene.