17 THE SEMIALGEBRAIC CASE Today’s Goal Tarski-Seidenberg Theorem. For every n =1, 2, 3,...,every Lalg-definable subset of Rn can be defined by a quantifier-free Lalg-formula. n Thus every Lalg-definable subset of R is a finite boolean combination (i.e., finitely many intersections, unions, and comple- ments) of sets of the form n {(x1,...,xn) ∈ R | p(x1,...,xn) > 0} where p(x1,...,xn)isapolynomial with coefficients in R. These are called the semialgebraic sets. A function f: A ⊂ Rn → Rm is semialgebraic if its graph is a semialgebraic subset of Rn × Rm. 18 As for the semilinear sets, every semialgebraic set can be written as a finite union of the intersection of finitely many sets defined by conditions of the form p(x1,...,xn)=0 q(x1,...,xn) > 0 where p(x1,...,xn) and q(x1,...,xn) are polynomials with coefficients in R. Proof Strategy • Prove a geometric structure theorem that shows that any semialgebraic set can be decomposed into finitely many semialgebraic generalized cylinders and graphs. • Deduce quantifier elimination from this. 19 Thom’s Lemma. Let p1(X),...,pk(X) be polynomials in the variable X with R coefficients in such that if pj(X) =0then pj(X) is included among p1,...,pk. Let S ⊂ R have the form S = ∩jpj(X) ∗j 0 where ∗j is one of <, >,or=, then S is either empty, a point, or an open interval. Moreover, the (topological) closure of S is obtained by changing relaxing the sign conditions (changing < to ≤ and > to ≥. Note There are 3k such possible sets, and these form a partition of R. 20 Some Standard Tricks • Identify the complex numbers C with R2 via a + bi ←→ (a, b) √ where a, b ∈ R and i = −1. With this identification, multiplication of complex numbers is a semialgebraic function from R2 × R2 to R2. Also, Cn is identified with R2n • The collection of polynomials in the variable X with coefficients in R of degree not greater than n can be identified with Rn+1 via n a0+a1X+···+anX ↔ (a0,a1,...,an). Similarly for polynomials with coeffi- cients in C. Addition, multiplication, differentiation of polynomials are semi- algebraic functions. 21 n R Rn+1 Let Bk ( ) denote (as a subset of ) the collection of polynomials in the variable X with real coefficients of degree not greater than n that have exactly k distinct complex roots. n R ⊂ n R Let Mk ( ) Bk ( )bethose polynomials of degree n with this property. ⊂ n R Lemma. Suppose that A Mk ( ) is connected. For each a¯ ∈ A let ra¯ be the number of distinct real roots of the polynomial pa¯(X) associated with a¯. Then a. ra¯ = r is constant on A; b. There are continuous functions f1,...,fr: A → R such that for all a¯ ∈ A we have fi(¯a) <fi+1(¯a) for i =1,...,r− 1 and pa¯(fi(¯a))=0for i =1,...,r. (“Continuity of roots”) 22 n R n R Lemma. The subsets Bk ( ) and Mk ( ) of Rn+1 are semialgebraic. Idea • The polynomial p(X) has a repeated root if and only if p(X) and its derivative p(X)haveacommon factor. • This can be expressed by the condition that the determinant of a matrix constructed from the coefficients of p(X), the so-called resultant of p and p (also called the discriminant of p), has value 0. • This is a semialgebraic condition on the coefficients. • Then extend this idea to capture n R n R Bk ( ) and Mk ( ) semialgebraically. 23 Graphs and Cylinders The structure theorem shows that a semialgebraic set S ⊆ Rn can be partitioned into finitely many sets of two kinds, all of which are semialgebraic. Graphs Let A ⊂ Rk and f: A → R be continuous. The graph of f is the subset of Rk+1 given by Graph(f)={(¯x, y) ∈ Rk+1 | x¯ ∈ A and y = f(¯x)}. Generalized Cylinders Let A ⊂ Rk, and let f,g: A → R be continuous and satisfy f(¯x) <g(¯x) for all x¯ ∈ A. The cylinder determined by f, g, and A is the subset of Rk+1 given by k+1 (f,g)A = {(¯x, y) ∈ R | x¯ ∈ A and f(¯x) <y<g(¯x)}. 24 • If A is connected, then graphs and cylinders based on A are connected. Structure Theorem. Let S ⊂ Rn be semialgebraic. Then: In. S has finitely many connected compo- nents and each one is semialgebraic n−1 IIn. There is a finite partition P of R into connected semialgebraic sets such that for each A ∈Pthere is kA ∈ N A → R ∪ {±∞} and fi : A for i = 0, 1 ...,kA +1satisfying a. f A = −∞, f A = ∞, f A is 0 kA+1 i continuous for 1 ≤ i ≤ kA, and A A ≤ ≤ fi−1(¯x) <fi (¯x) for all 1 i kA +1and x¯ ∈ A; A ≤ b. all graph sets Graph(fi ) for 1 i ≤ kA and generalized cylinders A A (fi−1,fi )A are semialgebraic. The graphs and cylinders in (b) for all A ∈P partitions Rn and S. 25 The Proof • The proof is by induction on n, and I shall outline the induction step. • Most broadly, the argument is as follows: show In−1 ⇒ IIn and IIn ⇒ In. • IIn ⇒ In is evident; the crux is In−1 ⇒ IIn. • Split the coordinates of Rn as (x1,...,xn−1,t). • Using standard set theory, write S as the union of finitely many finite intersections of polynomial equalities and inequalities. • Extend the finite collection of polyno- mials in the given definition of S by in- cluding all iterated partial derivatives with respect to t. Let this expanded list of polynomials be q1,...,qr. 26 • For each subset S⊂{1,...,r}, form the polynomial QS (¯x, t)= qj(¯x, t). j∈S Viewx ¯ as parameter variables and consider the polynomial as QS,x¯(t), apolynomial in the variable t whose coefficients are polynomials inx ¯. • For each <degree QS,x¯(t) and k ≤ , let { ∈ Rn−1 | MS,k = x¯ degree QS,x¯(t)= and it has exactly k distinct real roots}. It can be shown that MS,k is semialge- braic. • Partition Rn−1 by taking all intersec- tions of all MS,k. This still is a finite semialgebraic partition of Rn−1. 27 • Refine this partition further to obtain a partition P0 by taking the connected components of the sets in the partition above. By In−1 this again is a finite semialgebraic partition of Rn−1. • For A ∈P0, let QA,x¯(t)bethe product of those qj(¯x, t) which are nonzero for (all)x ¯ ∈ A.Itcan be shown that the number of roots of QA,x¯(t)isuniform asx ¯ ranges over A and that the 1st, 2nd, ... root functions are continuous on A,asA is connected. • Form the corresponding graph and generalized cylinder sets above each set A ∈P0. • It can be shown that each such set has the form ∩r { | ∈ ∗ } j=1 (¯x, t) x¯ A and qj(¯x, t) j 0 where ∗j is one of <, >,or=. This step uses Thom’s Lemma. 28 Tarski-Seidenberg Theorem redux. Let f: X ⊂ Rn → Rm be semialgebraic. Then the image of f, f(X)={y¯ ∈ Rm|y¯ = f(¯x) for some x¯ ∈ X}, is semialgebraic. 29 Algorithmic Cruelty Tarski’s original proof gives an algo- rithm for quantifier-elimination: given an Lalg-formula as input, it outputs a quanti- fier-free formula that defines the same set as the input formula. Computationial efficiency of a quanti- fier elimination algorithm thus becomes im- portant for applications (e.g., robot motion planning). Cylindrical algebraic decomposi- tion-based quantifier elimination, such as described above and developed in 1973 by Collins, has played an important role. Quantifier elimination for Ralg is, un- fortunately, an inherently computationally intensive problem. It is known that there is a doubly exponential lower bound in the number of quantifiers for worst-case time complexity. 30 Some References R. Benedetti and J.-J. Risler, Real algebraic and semi-algebraic sets,Paris: Hermann, 1990. (particularly Chapters 1 and 2) J. Bochnak, M. Coste, and M.-F. Roy, Real algebraic geometry, Berlin: Springer- Verlag, 1998. B.F. Caviness and J. R. Johnson (eds.), Quantifier Elimination and Cylindrical Al- gebraic Decomposition, Vienna: Springer- Verlag, 1998. L. van den Dries, Tame Topology and O- minimal Structures (London Mathematical Society Lecture Note Series, vol. 248), Cambridge: Cambridge University Press, 1998. (Chapter 2).