On predicates in algebraically closed fields

E-ISSN： 1943-5886|19|2|103-114

ISSN： 0022-4812

Source： The Journal of Symbolic Logic, Vol.19, Iss.2, 1954-06, pp. : 103-114

Disclaimer: Any content in publications that violate the sovereignty, the constitution or regulations of the PRC is not accepted or approved by CNPIEC.

Previous Menu Next

Abstract

Many properties of curves, surfaces, or other varieties in Algebraic Geometry can be formulated in the lower functional calculus as predicates of the coefficients of the polynomial or polynomials which define the variety (curve, surface) in question. For example, the property of a plane curve of order n to possess exactly m double points, or the property to be of genus p — where m and p are specified integers — can be formulated in this way. Similarly many statements on the relation between two or more varieties, e.g., concerning the number and type of their intersection points, can be expressed in the lower functional calculus. It is usual to study the properties of a variety in an algebraically closed field. Accordingly, it is of considerable interest to investigate the general structure of the class of predicates mentioned above in relation to algebraically closed fields. The following result will be proved in the present paper.