Chapter VIII. Background for Algebraic Geometry

Abstract

This chapter introduces aspects of the algebraic theory of systems of polynomial equations in several variables.

Section 1 gives a brief history of the subject, treating it as one of two early sources of questions to be addressed in algebraic geometry.

Section 2 introduces the resultant as a tool for eliminating one of the variables in a system of two such equations. A first form of Bezout's Theorem is an application, saying that if $f(X,Y)$ and $g(X,Y)$ are polynomials of respective degrees $m$ and $n$ whose locus of common zeros has more than $mn$ points, then $f$ and $g$ have a nontrivial common factor. This version of the theorem may be regarded as pertaining to a pair of affine plane curves.

Section 3 passes to projective plane curves, which are nonconstant homogeneous polynomials in three variables, two such being regarded as the same if they are multiples of one another. Versions of the resultant and Bezout's Theorem are valid in this context, and two projective plane curves defined over an algebraically closed field always have a common zero.

Sections 4–5 introduce intersection multiplicity for projective plane curves. Section 4 treats a line and a curve, and Section 5 treats the general case of two curves. The theory in Section 4 is completely elementary, and a version of Bezout's Theorem is proved that says that a line and a curve of degree $d$ have exactly $d$ common zeros, provided the underlying field is algebraically closed, the zeros are counted as often as their intersection multiplicities, and the line does not divide the curve. Section 5 makes more serious use of algebraic background, particularly localizations and the Nullstellensatz. It gives an indication that ostensibly simple phenomena in the subject can require sophisticated tools to analyze.

Section 6 proves a version of Bezout's Theorem appropriate for the context of Section 5: if $F$ and $G$ are two projective plane curves of respective degrees $m$ and $n$ over an algebraically closed field, then either they have a nontrivial common factor or they have exactly $mn$ common zeros when the intersection multiplicities of the zeros are taken into account.

Sections 7–10 concern Gröbner bases, which are finite generating sets of a special kind for ideals in a polynomial algebra over a field. Section 7 sets the stage, introducing monomial orders and defining Gröbner bases. Section 8 establishes a several-variable analog of the division algorithm for polynomials in one variable and derives from it a usable criterion for a finite set of generators to be a Gröbner basis. From this it is easy to give a constructive proof of the existence of Gröbner bases and to obtain as consequences solutions of the ideal-membership problem and the proper-ideal problem. Section 9 obtains a uniqueness theorem under the condition that the Gröbner basis be reduced. Adjusting a Gröbner basis to make it reduced is an easy matter. A consequence of the uniqueness result is a solution of the ideal-equality problem. Section 10 gives two theorems concerning solutions of systems of polynomial equations. The Elimination Theorem identifies in terms of Gröbner bases those members of the ideal that depend only on a certain subset of the variables. The Extension Theorem, proved under the additional assumption that the underlying field is algebraically closed, gives conditions under which a solution to the subsystem of equations that depend on all but one variable can be extended to a solution of the whole system. The latter theorem makes use of the theory of resultants.