On predicates in algebraically closed fields

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 ordernto possess exactlymdouble points, or the property to be of genusp— wheremandpare 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.Main Theorem.LetFbe a commutative algebraic field of arbitrary characteristic, and letF′ = F[x₁, …,x_n]be the ring of polynomials of n variables with coefficients inF.With every predicate Q{x₁, …,x_n)which is formulated in the lower functional calculus in terms of the relations of equality, addition, and multiplication and (possibly) in terms of some of the elements ofF,there can be associated an ascending chain of ideals inF′, such that for every extensionF*ofFwhich is algebraically closed. In this formula, V₀, …, V_{2k+ 1}are the varieties of the ideals J₁, …, J_2k+iin the coordinate spaceS_n: (x₁, …x_n) over F*, and V_Qis the set of points ofS_nwhich satisfyQ.