An introduction to semialgebraic geometry

78 Pages·2002·1.124 MB·English

by  Coste M.

AN INTRODUCTION
TO SEMIALGEBRAIC GEOMETRY
Michel COSTE
Institut de Recherche Math´ematique de Rennes
[e-mail]
Octobre 2002
Contents
Introduction 3
1 Counting the real roots 5
1.1 Sturm’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.1.1 P without multiple root . . . . . . . . . . . . . . . . . . 5
1.1.2 P with multiple roots . . . . . . . . . . . . . . . . . . . . 6
1.1.3 A bound for the roots . . . . . . . . . . . . . . . . . . . 7
1.2 Real roots satisfying inequalities . . . . . . . . . . . . . . . . . . 7
1.2.1 One inequality . . . . . . . . . . . . . . . . . . . . . . . 7
1.2.2 Several inequalities . . . . . . . . . . . . . . . . . . . . . 9
1.2.3 Deciding the existence of a solution of a system of poly-
nomial equations and inequalities . . . . . . . . . . . . . 10
1.3 Systems with parameters . . . . . . . . . . . . . . . . . . . . . . 10
1.3.1 Tarski-Seidenberg . . . . . . . . . . . . . . . . . . . . . . 11
1.3.2 Systems with one equation . . . . . . . . . . . . . . . . . 12
1.3.3 Example: polynomial of degree 4 . . . . . . . . . . . . . 13
1.3.4 General systems . . . . . . . . . . . . . . . . . . . . . . . 16
1.4 Another method for counting real roots . . . . . . . . . . . . . . 17
1.4.1 Hermite’s method . . . . . . . . . . . . . . . . . . . . . . 17
1.4.2 Determination of the signature . . . . . . . . . . . . . . 19
1.4.3 Principal subresultant coefficients . . . . . . . . . . . . . 21
2 Semialgebraic sets 25
2.1 Stability properties . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.1.1 Definition and first examples . . . . . . . . . . . . . . . . 25
2.1.2 Consequences of Tarski-Seidenberg principle . . . . . . . 26
2.2 Semialgebraic functions . . . . . . . . . . . . . . . . . . . . . . . 29
2.2.1 Definition and first properties . . . . . . . . . . . . . . . 29
2.2.2 The L8 ojasiewicz inequality . . . . . . . . . . . . . . . . . 30
1
2 CONTENTS
2.3 Decomposition of a semialgebraic set . . . . . . . . . . . . . . . 32
2.3.1 Cylindrical algebraic decomposition . . . . . . . . . . . . 32
2.3.2 Construction of an adapted c.a.d. . . . . . . . . . . . . . 34
2.3.3 The c.a.d. algorithm . . . . . . . . . . . . . . . . . . . . 38
2.3.4 Connected components of semialgebraic sets . . . . . . . 40
3 Triangulation of semialgebraic sets 41
3.1 Thom’s lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.1.1 For one variable . . . . . . . . . . . . . . . . . . . . . . . 43
3.1.2 For several variables . . . . . . . . . . . . . . . . . . . . 44
3.1.3 The finiteness theorem . . . . . . . . . . . . . . . . . . . 48
3.2 Triangulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.2.1 Simplicial complexes . . . . . . . . . . . . . . . . . . . . 49
3.2.2 Triangulation of a compact semialgebraic set . . . . . . . 51
3.2.3 The curve selection lemma . . . . . . . . . . . . . . . . . 54
3.3 Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.3.1 Dimension via c.a.d. . . . . . . . . . . . . . . . . . . . . 55
3.3.2 Dimension of algebraic sets . . . . . . . . . . . . . . . . . 57
4 Families. Uniform bounds 61
4.1 Semialgebraic triviality of families . . . . . . . . . . . . . . . . . 61
4.1.1 Hardt’s theorem . . . . . . . . . . . . . . . . . . . . . . . 61
4.1.2 Local conic structure of semialgebraic sets . . . . . . . . 63
4.1.3 Finiteness of the number of topological types . . . . . . . 64
4.2 Number of connected components . . . . . . . . . . . . . . . . . 65
4.2.1 Reduction to the case of a compact smooth hypersurface 67
4.2.2 The case of a compact smooth hypersurface . . . . . . . 68
4.2.3 Bound for the number of connected components of a
semialgebraic set . . . . . . . . . . . . . . . . . . . . . . 70
4.3 An application to lower bounds . . . . . . . . . . . . . . . . . . 70
Bibliography 75
Index 77
Introduction
Semialgebraic geometry is the study of sets of real solutions of systems of
polynomial equations and inequalities. These notes present the first results of
semialgebraic geometry and related algorithmic issues. Their content is by no
means original.
The first chapter explains algorithms for counting real roots (this is, in
some sense, 0-dimensional semialgebraic geometry) and the Tarski-Seidenberg
theorem.
In the next two chapters we study semialgebraic subsets of Rn, which are
definedbybooleancombinationsofpolynomialequationsandinequalities. The
main tool for this study is the cylindrical algebraic decomposition, which is
introduced in Chapter 2. The principal result of Chapter 3 is the triangulation
theorem. This theorem shows that semialgebraic sets have a simple topology,
which can be effectively computed from their definitions.
Chapter 4 contains examples of finiteness results and uniform bounds for
semialgebraic families. In particular, we give an explicit bound for the number
of connected components of a real algebraic set as a function of the degree of
the equations and the dimension of the ambient space.
Many results of semialgebraic geometry also hold true for o-minimal struc-
tures (including for instance classes of sets definable with the exponential func-
tions). See [D] (or the lecture notes [Co]) for this theory. The algorithmic
aspects are specific to the semialgebraic case.
The bibliography is reduced to a minimum, containing mainly books and
recent surveys. Further references (in particular, references to original papers)
may be found there.
These notes have served as a basis for courses in Rennes, Paris and Pisa, for
mini-courses at MSRI (Berkeley), at a summer school in Laredo (Cantabria)
and at a CIMPA school in Niamey.
3
4 INTRODUCTION
Chapter 1
Counting the real roots of a
polynomial
1.1 Sturm’s theorem
In this section, P ∈ R[X] is a nonconstant polynomial in one variable.
1.1.1 P without multiple root
(cid:1)
We assume that P has no multiple root, i.e., gcd(P,P ) = 1. We construct
(cid:1)
a sequence of polynomials in the following way: P = P, P = P and, for
0 1
i > 0, Pi+1 is the negative of the remainder of the euclidean division of Pi−1
by Pi (Pi−1 = PiQi − Pi+1, with degPi+1 < degPi). We stop just before we
get 0. The last polynomial P is then a nonzero constant (up to signs, this is
K
just Euclide’s algorithm for computing the gcd). The sequence P ,...,P is
0 K
called the Sturm sequence of P and P(cid:1). Let a ∈ R, not a root of P. Denote
by v (a) the number of sign changes in the sequence P (a),P (a),...,P (a).
P 0 1 K
For instance, if P = X3 −3X +1, the Sturm sequence of P and P(cid:1) is
9
(X3 −3X +1, 3X2 −3, 2X −1, ) ,
4
which gives (−1,0,1, 9) by evaluation at a = 1. Here v (1) = 1. We drop the
4 P
zeroeswhichoccurwhencountingthesignchanges. Theresultisthefollowing:
Theorem 1.1 (Sturm) Let a < b in R, neither a nor b being a root of P.
The number of roots of P in the interval (a,b) is equal to v (a)−v (b).
P P
5
6 CHAPTER 1. COUNTING THE REAL ROOTS
Proof. We consider how v (x) changes when x passes through a root c of a
P
polynomial of the Sturm sequence.
• If c is a root of P, the signs of P and P behave as follows:
0 1
x c x c
either P − 0 + or P + 0 − .
0 0
P + + + P − − −
1 1
In both cases, the contribution to v (x) decreases by 1.
P
• If c is a root of Pi, 0 < i < K, we have Pi−1(c) = −Pi+1(c) (cid:4)= 0. Hence
the contribution of the subsequence Pi−1(x),Pi(x),Pi+1(x) to vP(x) does
not change and remains equal to 1.
The theorem follows from the preceding remarks. (cid:5)(cid:6)
1.1.2 P with multiple roots
The proof of Theorem 1.1 relies on the following properties of the sequence
P ,...,P :
0 K
1. P = P , and P is a nonzero constant.
0 K
2. If c is a root of P , the product P P is negative on some interval (c−ε,c)
0 0 1
and positive on some interval (c,c+ε).
3. If c is a root of Pi, 0 < i < K, then Pi−1(c)Pi+1(c) < 0.
Assume now that P has multiple roots. We construct, as above, the sequence
(cid:1)
P = P,P = P ,...,P . Now P is no longer a constant, but the gcd of P
0 1 K K
(cid:1)
and P . Consider the sequence
P0/PK, P1/PK,..., PK−1/PK, 1 .
This sequence satisfies properties 1-2-3 above for the polynomial P /P , which
0 K
has the same roots as P (not counting multiplicities). Moreover, if a is not a
root of P, the number v (a) of sign changes in the sequence P (a),...,P (a)
P 0 K
is obviously the same as the number of sign changes in the sequence
P0(a)/PK(a),..., PK−1(a)/PK(a), 1 .
It follows:
Theorem 1.2 Sturm’s theorem still holds if P has multiple roots. The differ-
ence v (a)−v (b) is equal to the number of distinct roots of P in the interval
P P
(a,b).
1.2. REAL ROOTS SATISFYING INEQUALITIES 7
1.1.3 A bound for the roots
Proposition 1.3 Let P = a0Xd +···+ad−1X +ad, where a0 (cid:4)= 0. If c ∈ C
is a root of P, then
(cid:2) (cid:3) (cid:3)(cid:4)
(cid:3)a (cid:3) 1/i
|c| ≤ max d (cid:3)(cid:3) i(cid:3)(cid:3) .
i=1,...,d a0
Proof. Set M = max (d |a /a |)1/i and let z ∈ C be such that |z| > M.
i=1,...,d i 0
Then |a | < |a ||z|i/d, for i = 1,...,d. Hence,
i 0
|a zd−1 +···+a | ≤ |a ||z|d−1 +···+|a | < |a zd|
1 d 1 d 0
and P(z) (cid:4)= 0. (cid:5)(cid:6)
Set
(cid:2) (cid:3) (cid:3)(cid:4)
(cid:3)a (cid:3) 1/i
(cid:3) i(cid:3)
M = max d (cid:3) (cid:3) .
i=1,...,d a0
Then v (x) is constant on (−∞,−M) (resp. (M,+∞)) and equal to v (−∞)
P P
(resp. v (+∞)), whichisthenumberofsignchangesinthesequenceofleading
P
coefficients of P (−X), P (−X), ..., P (−X) (resp. P (X), ..., P (X)).
0 1 K 0 K
Proposition 1.4 The total number of distinct real roots of P is
v (−∞)−v (+∞).
P P
1.2 Real roots satisfying inequalities
In this section, P ∈ R[X] is a non constant polynomial in one variable, and
Q, Q , ..., Q are polynomials in R[X].
1 (cid:9)
1.2.1 One inequality
We want to count the number of real roots c of P such that Q(c) > 0. We
(cid:1)
modify the construction of the Sturm sequence by taking P = P, P = P Q
0 1
and, as before, P = the negative of the remainder of the euclidean division
i+1
of Pi−1 by Pi, for i > 0. We stop just before we obtain 0, i.e. we stop with PK
(cid:1)
which is the gcd of P and P Q. The sequence of polynomials we obtain in this
(cid:1)
way is called the Sturm sequence of P and P Q. If the real number a is not
a root of P, we denote by v (a) the number of sign changes in the sequence
P,Q
P (a),P (a),...,P (a).
0 1 K
8 CHAPTER 1. COUNTING THE REAL ROOTS
Theorem 1.5 Let a < b be real numbers which are not roots of P. Then
v (a)−v (b) is equal to the number of distinct roots c of P in (a,b) such
P,Q P,Q
that Q(c) > 0 minus the number of those such that Q(c) < 0.
(cid:1)
Proof. First consider the case where P and P Q are relatively prime (P is a
K
nonzero constant). This means that P has no multiple root, and no common
root with Q. The property 2 of Section 1.1.2 is replaced with:
2’ If c is a root of P , the product P P Q is negative on some interval
0 0 1
(c−ε,c) and positive on some interval (c,c+ε).
The theorem follows from properties 1-2’-3.
If PK is not a constant, the sequence P0/PK,P1/PK,...,PK−1/PK,1 sat-
isfies properties 1-2’-3 for P /P . Hence, the difference between the numbers
0 K
of sign changes in this sequence evaluated at a and b, respectively, is equal
to the number we want to calculate, and it coincides with v (a) − v (b).
P,Q P,Q
(cid:5)(cid:6)
Remark. v (a)−v (b) counts the number of distinct roots of P in (a,b)
P,Q2 P,Q2
which are not real roots of Q. Therefore the number of distinct roots c of P
in (a,b) such that Q(c) > 0 is equal to
1
(v (a)+v (a)−v (b)−v (b)) .
P,Q P,Q2 P,Q P,Q2
2
We can replace v with v if P and Q are relatively prime.
P,Q2 P
Exercise 1.6 The Cauchy index of a rational fraction F ∈ R(X) between a
andbisthenumberofpolesc,witha < c < b,suchthatlimx→c−F(x) =
−∞ and limx→c+F(x) = +∞, minus the number of those such that
limx→c−F(x) = +∞ and limx→c+F(x) = −∞.
Given two polynomials P and Q in R[X], define the Sturm sequence
of P and Q by taking P = P, P = Q and the rest constructed as
0 1
above. If a and b are not roots of P, show that the Cauchy index of
Q/P between a and b is equal to the difference v(a)−v(b) between the
numbers of sign changes in the Sturm sequence evaluated at a and b,
respectively. Recover Theorems 1.1 and 1.5 from this result.
Exercise 1.7 Show that we get the same result as Theorem 1.5 if we replace
(cid:1) (cid:1)
P Q with the remainder of the euclidean division of P Q by P in the
construction of the Sturm sequence.
1.2. REAL ROOTS SATISFYING INEQUALITIES 9
1.2.2 Several inequalities
WewanttocountthenumberofrealrootsofasystemP = 0, Q > 0,..., Q >
1 (cid:9)
0. First assume that P is relatively prime with all Q . Let ε = (ε ,...,ε ) ∈
i 1 (cid:9)
{0,1}(cid:9) and Qε = Qε1···Qε(cid:1). By Theorem 1.5, s = v (−∞)−v (+∞) is
1 (cid:9) ε P,Qε P,Qε
equal to the number of distinct real roots c of P such that Qε(c) > 0 minus the
number of those such that Qε(c) < 0. If ϕ = (ϕ ,...,ϕ ) ∈ {0,1}(cid:9), we denote
1 (cid:9)
by c the number of distinct real roots c of P such that the sign of Q (c) is
ϕ i
(−1)ϕi, for i = 1,...,(cid:23). Let s (resp. c) be the vector whose coordinates are all
s (resp. c ).
ε ϕ
Lemma 1.8 There is an invertible 2(cid:9) × 2(cid:9) matrix A , depending only on (cid:23),
(cid:9)
such that s = A .c.
(cid:9)
Proof. We proceed by induction on (cid:23). For (cid:23) = 0, we have trivially s = c .
∅ ∅
For (cid:23) = 1, Sturm’s theorem and Theorem 1.5 imply
(cid:5) (cid:6) (cid:5) (cid:6)(cid:5) (cid:6)
s 1 1 c
0 = 0 .
s 1 −1 c
1 1
The induction step from (cid:23) to (cid:23)+1 is as follows:
   
. .
. .
. .
   
   
 s  (cid:5) (cid:6) c 
 ε...,0  = AA(cid:9) −AA(cid:9)  ϕ...,0  .
 s  (cid:9) (cid:9)  c 
 ε,1   ϕ,1 
. .
. .
. .
(cid:5) (cid:6)
A A
(Exercise: check this equality). The matrix A = (cid:9) (cid:9) has inverse
(cid:9)+1 A −A
(cid:9) (cid:9)
(cid:5) (cid:6)
1 A−1 A−1
(cid:9) (cid:9) .
2 A−1 −A−1
(cid:9) (cid:9)
(cid:5)(cid:6)
We obtain from the lemma c = A−1.s. Since we can compute s, we get c
(cid:9)
and, in particular, the number of solutions of P = 0,Q > 0,...,Q > 0. In the
1 (cid:9)
general case (P can have multiple roots or common roots with Q ), we replace
(cid:13)(cid:14) (cid:15) i
Qε with (cid:9) Q2 /Qε, in order to get rid of the roots of P which are also
i=1 i
roots of some Q .
i