Chapter 2: Basic Real Algebraic Geometry Adam Sheffer April 15, 2015 “Every field has its taboos. In algebraic geometry the taboos are (1) writing a draft that can be followed by anyone but two or three of one’s closest friends, (2) claiming that a result has applications, (3) mentioning the word “combinatorial”, and (4) claiming that algebraic geometry existed before Grothendieck.” / “Indiscrete Thoughts” by Gian-Carlo Rota. In this chapter, we introduce very basic algebraic geometry over the reals. People with previous background in algebraic geometry would probably be rather bored with definitions of ideals, varieties, etc. However, they may be surprised to learn how nonintuitive things are when working over the reals (especially in the second half of this chapter). Although later on we may consider C and Fp as our fields, at this point we only consider R. 1 Ideals and varieties Algebraic geometry can be thought of as the study of geometries that arise from algebra (or more specifically, from polynomials). In this section we present varieties, which are the basic geometric object of algebraic geometry, and ideals, which are the main algebraic objects that we study in this course. The polynomial ring R[x1,...,xd] is the set of all polynomials in x1,...,xd with co- efficients in R. Given a (possibly infinite) set of polynomials f1,...,fk ∈ R[x1,...,xd], the affine variety V(f1,...,fk) is defined as d V(f1,...,fk)= {(a1,...,ad) ∈ R : fi(a1,...,ad) = 0 for all 1 ≤ i ≤ k}. The adjective “affine” distinguishes the variety from projective varieties. At this point we only consider affine varieties, and for brevity refer to those simply as vari- eties.1 For example, some varieties in R3 are a torus, the union of a circle and a line, and a set of 1000 points. The following is a special case of Hilbert’s basis theorem (e.g., see [2, Section 2.5]). Theorem 1.1. Every variety can be described by a finite set of polynomials. 1To make things even worse, some authors call these objects algebraic sets, while using the word variety for what we will refer to as an irreducible variety. 1 Theorem 1.1 is valid in every field. When working over the reals, we can say something stronger. Corollary 1.2. Every variety can be described by a single polynomial. d Proof. Consider a variety U ⊂ R . By Theorem 1.1, there exist f1,...,fk, ∈ R[x1,...,xd] 2 2 2 such that U = V(f1,...,fk). We set f = f1 + f2 + ··· + fk . Notice that for any d point p ∈ R we have f(p) = 0 if and only if f1(p)= ··· = fk(p) = 0. Thus, we have U = V(f). We consider some basic properties of varieties. Claim 1.3. Let U, W ⊂ Rd be two varieties, and let τ : Rd → Rd be an invertible linear map (such as a translation, rotation, reflection, stretching, etc.). Then (a) U ∩ W is a variety, (b) U ∪ W is a variety, and (c) τ(U) is a variety. Proof. Since U and W are varieties, there exist f1,...,fk,g1,...,gm ∈ R[x1,...,xd] 2 such that U = V(f1,...,fk) and W = V(g1,...,gm). For (a), notice that we have U ∩ W = V(f1,...,fk,g1,...,gm). For (b), we have U ∪ W = V(H), where H = {fi · gj}. 1≤i≤k 1≤[j≤m d For (c), we write the inverse of τ as ψ ∈ (R[x1,...,xd]) . Then τ(U)= V (f1 ◦ ψ,...,fd ◦ ψ) . At this point it might be instructive to ask what subsets of Rd are not varieties. Claim 1.4. The set X = {(x, x): x ∈ R, x =06 } ⊂ R2 is not a variety. Proof. Assume for contradiction that there exist f1,...,fk ∈ R[x1, x2] such that X = V(f1,...,fk). For every 1 ≤ i ≤ k, we set gi(t) = fi(t, t) and notice that gi ∈ R[t]. Since by definition gi(t) vanishes on every t =6 1, we have that gi(t) = 0. This in turn implies that fi(1, 1) = 0. Since this holds for every 1 ≤ i ≤ k, we get a contradiction to (1, 1) ∈/ X. Similarly, a line segment and half a circle are not varieties. We say that a set U ′ is a subvariety of a variety U if U ′ ⊆ U and U ′ is a variety. We say that U ′ is a proper subvariety if U ′ =6 ∅, U. A variety U is reducible if there exist two proper subvarieties U ′, U ′′ ⊂ U such that U = U ′ ∪ U ′′. Otherwise, U is irreducible. For example, the union of the two axes V(x, y) ⊂ R2 is reducible since V(x, y)= V(x) V(y). 2By Corollary 1.2,S it suffices to use a single polynomial for each variety. We present this slightly less elegant proof since it applies in every field. 2 Every variety U can be decomposed into irreducible subvarieties U1,...,Uk such k that U = i=1 Uk. After removing any Ui that is a proper subvariety of another Uj, we obtain a unique decomposition of U. The subvarieties of this decomposition are said to beS the irreducible components of U (or components, for brevity). Ideals. A subset J ⊆ R[x1,...,xd] is an ideal if it satisfies: • 0 ∈ J. • If f,g ∈ J then f + g ∈ J. • If f ∈ J and h ∈ R[x1,...,xd], then f · h ∈ J. As a first example of an ideal, consider a polynomial f ∈ R[x1,...,xd] and no- tice that {f · h : h ∈ R[x1,...,xd]} is an ideal. More generally, given f1,...,fk ∈ R[x1,...,xd], the set k hf1,...,fki = fi · hi : h1,...,hk ∈ R[x1,...,xd] ( i=1 ) X 3 is an ideal. We say that this ideal is generated by f1,...,fk. We also say that {f1,...,fk} is a basis of this ideal. We are specifically interested in ideals of varieties. Given a variety U ⊂ Rd, the ideal of U is I(U)= {f ∈ R[x1,...,xd]: f(a) = 0 for every a ∈ V }. It can be easily verified that I(U) satisfies the three requirements in the definition of an ideal. This seems to lead to the following question: Given f1,...,fk ∈ R[x1,...,xd], is it always the case that hf1,...,fki = I(V(f1,...,fk)) ? Claim 1.5. Given f1,...,fk ∈ R[x1,...,xd], we have hf1,...,fki ⊆ I(V(f1,...,fk)) although equality need not occur. Proof. To see that the containment relation holds, we set U = V(f1,...,fk). If g ∈ hf1,...,fki then by definition g vanishes on every point of U, and is thus in I(V(f1,...,fk)). To see that equality does not always hold, we set f = x2 + y2. We then have V(f)= {(0, 0)} ⊂ R2, even though x ∈ I(V(f)) and x∈h / fi. When defining a variety U, it is often useful to use a basis of I(U) rather than an arbitrary set of polynomials that define U. We conclude this section by inspecting another connection between ideals and varieties. Claim 1.6. Let U, W ⊂ Rd be varieties. Then (a) U ⊂ W if and only if I(W ) ⊂ I(U). (b) U = W if and only if I(W )= I(U). 3There are many different notations for an ideal generated by a set of polynomials. We use h·i following the notation of [2]. 3 Proof. First assume that U ⊂ W and let f ∈ I(W ). That is, the polynomial f vanishes on every point of W . Since U ⊂ W , then f vanishes on every point of U, which implies f ∈ I(U). To see that this containment is proper, notice that there must exist polynomials that vanish on U but not on W (otherwise we would have U = W ). Next, assume that I(W ) ⊂ I(U) and let p ∈ U. Every polynomial of I(U) vanishes on the point p, which in turn implies that every polynomial of I(W ) vanishes on p. Thus, we have p ∈ W . Part (b) is proved in a similar manner. 2 Dimension Consider an irreducible variety U ⊂ Rd. One intuitive definition of the dimension d′ of U, denoted dim U, is the maximum integer such that there exist U0 ⊂ U1 ⊂···⊂ Ud′ = U, where all of the subsets are proper and all of the sets Ui are irreducible varieties. This definition immediately implies that the intersection of two distinct irreducible varieties of dimension d′ is of dimension smaller than d′. d If U ⊂ R is a reducible variety with irreducible components U1,...,Uk, then the dimension of U is maxi{dim Ui}. A curve is a variety with all of its components of dimension one. A k-flat is a translation of a k-dimensional linear space. A hypersurface in Rd is a variety with all of its components of dimension d − 1. Similarly, a hyperplane in Rd is a (d − 1)-flat, a hypersphere is a (d − 1)-dimensional sphere, etc. We present the following claim without proof. d Claim 2.1. For every hypersurface U ⊂ R there exists a polynomial f ∈ R[x1,...,xd] such that hfi = I(U). Given an irreducible variety U ⊂ Rd of dimension d′, one might expect U to “look” like a d′-dimensional set in a small neighborhood around any point p ∈ U. To see that this is not the case, consider the cubic curve V(y2 −x3 +x2). Even though this is an irreducible variety, it consists of a one-dimensional curve together with the origin (see the left part of Figure 1).