» Bochnak J., Coste M., Roy M-F., Ames W.F. - Real algebraic geometry
Bochnak J., Coste M., Roy M-F., Ames W.F. - Real algebraic geometry
||Real algebraic geometry
||Bochnak J., Coste M., Roy M-F., Ames W.F.
||In simplest terms, algebraic geometry is the study of the set of solutions of a system of polynomial equations. The main goal of real algebraic geometry is the study of real algebraic sets i.e. subsets of Kn denned by polynomial equations. By means of a simple example one can see some features which point up the difference between real and complex algebraic geometry. Let us consider the intersection of the straight line x = t, depending on the parameter t, with the cubic y2 = x3 - x. For t = —1,0,1 the straight line is tangent to the cubic. In the complex plane, when t is different from -1,0,1, the straight line always intersects the cubic in two points. In the real plane the situation is more intricate.
a) The intersection may be empty because the field of real numbers is not algebraically closed. At first glance this appears to be a glaring defect. Just the same, this defect has some positive aspects, as we shall see.
b) While the intersection may be empty there is, nevertheless, an invariant, namely, the parity of the number of intersections (when the straight line is not tangent). This result can be traced back to complex conjugation, and for this reason one frequently encounters in real algebraic geometry invariants modulo 2.
c) The set of parameters t for which there is nonempty intersection is the union of two intervals. This set cannot be described by means of only polynomial equations and their negations (^). One has to make use of inequalities (viz. x3 - x > 0). One is thus led to consider semi-algebraic sets, which are the subsets of Rn defined by a finite number of polynomial equations and inequalities. The ordering of the reals, which plays an important role in this example, is also closely related to the euclidean topology of Rn, and is much more important for some real phenomena than the Zariski topology.
As the above example shows, real algebraic geometry is concerned not only with the zeros of polynomials but also with domains where the polynomials have constant sign. A famous example of this type is the 17th problem of Hilbert, which asks whether a polynomial which is nonnegative on Rn is a sum of squares of rational functions. The solution of this problem given by Artin has a remarkable feature: one is not able to resolve this question by considering only points of Rn. One has to consider points belonging to other fields, which contain the field of rational functions R(Xi,....,Xn) and have