Algebraic and Geometric Foundations
In the first part of the book we dealt exclusively with polyhedral and hence linear structures. Many situations explicitly or implicitly required computing the intersection of a finite set of affine hyperplanes in the n-dimensional space ℝ n . This was po
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Algebraic and Geometric Foundations
In the first part of the book we dealt exclusively with polyhedral and hence linear structures. Many situations explicitly or implicitly required computing the intersection of a finite set of affine hyperplanes in the n-dimensional space Rn . This was possible with methods from linear algebra. Although it is adequate to use linear geometric structures in many applications, there are also problems which have a natural non-linear representation. We restrict ourselves here to non-linear structures that can be handled with algebraic methods. This chapter is devoted to systems of polynomial equations in two unknowns.
8.1 Motivation So far we have mainly used the real numbers as a coordinate field. It would seem coherent, therefore, to continue this approach when we begin to look at non-linear geometry. However, it quickly becomes clear that algebraic geometry over the field R is significantly harder than over its algebraic closure C. Some of our results will be applicable for any field, some will focus exclusively on the complex numbers, and occasionally we will be able to transfer results from the complex to the real numbers. We begin by studying polynomials in one variable. The roots of a quadratic polynomial f (x) = x 2 + bx + c with real coefficients b, c can be real or complex. Regardless of the root type, we can express them in terms of radicals: b2 b − c. x1,2 = − ± 2 4 Similarly, for polynomials f of degree three or four there exist the so-called Cardano formulas,1 which provide explicit expressions for the zeros of f . 1 These
formulas are implemented in most computer algebra systems such as Maple and Sage.
M. Joswig, T. Theobald, Polyhedral and Algebraic Methods in Computational Geometry, Universitext, DOI 10.1007/978-1-4471-4817-3_8, © Springer-Verlag London 2013
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Algebraic and Geometric Foundations
Fig. 8.1 The graph of the function f (x) = x 5 − 4x + 2
The situation for polynomials of degree ≥ 5, however, is completely different. Galois theory shows that the zeros of a polynomial of degree ≥ 5 are in general not expressible in terms of radicals. An example of such a polynomial (which has the symmetric group of degree 5 as its Galois group) is x 5 − 4x + 2. So we might have to accept that we have to use a polynomial itself to “code” the zeros or to approximate the zeros with numerical methods. In this case we can give an approximation to the complex zeros as −1.518512,
0.508499,
1.243596,
−0.116792 ± 1.438448i.
The corresponding real function is illustrated in Fig. 8.1. Since polynomial systems are an extremely powerful tool in mathematical modeling, it is a central task of geometry to study the sets of zeros of arbitrary polynomials as well as the intersections of those sets. From a computational point of view we focus on computing and manipulating these sets. Example 8.1 The set of zeros of a quadratic polynomial f ∈ R[x, y] defines a conic section (or it is empty). For example, the polynomials f (x, y) = x 2 + y 2 − xy − x − y − 1, g(x, y) = 2x
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