An Application of Groebner Bases to Planarity of Intersection of Surfaces

In this paper we use Groebner bases theory with the purpose to determine planarity of intersections of two algebraic surfaces in R. There are specially considered plane sections of a type of conoid which has a cubic egg curve as one of the directrices. The paper investigates a possibility of conic plane sections of this type of conoid. 1. THE BASIC CONCEPT OF GROEBNER BASES Many problems in mathematics can be treated as a problem of solvability of a system of polynomial equations: (1) f1(x1, x2, . . . , xn) = 0 f2(x1, x2, . . . , xn) = 0 .. fm(x1, x2, . . . , xn) = 0 where f1, f2, . . . , fm are polynomials in n variables over set C. The basic topic in the theory of systems of polynomial equations are Groebner bases. Let us define this concept in the next consideration. The polynomial of several variables: (2) P (x1, . . . , xn) = ∑ α∈A0 cα x α1 1 · · ·xn n is considered as a sum over n-tuples α = (α1, . . . , αn) from a finite set A0⊂Nn0 (cα 6=0). The terms x1 1 · · ·xαn n are called the monomials, while the summands cα x α1 1 · · ·xαn n are called the terms of the polynomial. First author supported in part by the project MNTRS, Grant No. ON144020. 2000 Mathematics Subject Classifications. 14N05, 51N35.