Capítulo II. Marco Teórico
2.3. Teorías con otros enfoques
2.3.4. Roger Brown
Let d, m, n be positive integers such that we have n = d+m and d, m ≥ 1. Let x = x1 < · · · < xn be ordered variables, which are divided into two groups x1 < · · · < xd and xd+1 < · · · < xn. We rename xi as ui for 1 ≤ i ≤ d and see u =
u1, . . . , ud as parameters. We rename xi as yi−d for d + 1 ≤ i ≤ n and see y =
y1, . . . , ym as unknowns. In this section, we introduce the concept of acomprehensive
We first recall some notations on semi-algebraic systems introduced in the previous two chapters.
Semi-algebraic system. Let us consider four finite polynomial subsets F = {f1, . . . , fs}, N = {n1, . . . , nt}, P = {p1, . . . , pr} and H = {h1, . . . , hℓ} of
Q[x1, . . . , xn]. LetN≥ denote the set of the inequalities{n1 ≥0, . . . , nt ≥0}. Let P>
denote the set of the inequalities {p1 > 0, . . . , pr >0}. Let H6= denote the set of in- equations{h1 6= 0, . . . , hℓ 6= 0}. We denote byS= [F, N≥, P>, H6=] the semi-algebraic system which is the conjunction of the following conditions: f1 = 0, . . . , fs = 0,
n1 ≥ 0, . . . , nt ≥ 0, p1 > 0, . . . , pr > 0 and h1 6= 0, . . . , hℓ 6= 0. The system
f1 = 0, . . . , fs = 0, p1 6= 0, . . . , pr 6= 0 and h1 6= 0, . . . , hℓ 6= 0 is called the associated
constructible system of S. Its zero set in Cn is called the associated constructible set of S. We call [F,∅, P>, H6=], written as [F, P>, H6=] or [F, P>] when H6= is empty, a basic semi-algebraic system.
Squarefree semi-algebraic system. Let R := [T, P] be a squarefree regular system of Q[u,y]. We call the pairA:= [T, P>] a squarefree semi-algebraic system (SFSAS).
The system R is called theassociated regular system of A.
Definition 10.1. LetSbe a semi-algebraic system ofQ[u,y]. Letcsbe the associated constructible system ofS. A comprehensive triangular decompositionof S(RCTD)1 is a pair (C,(AC, C ∈ C)), where
C is a finite partition of R
d into nonempty semi-algebraic sets,
for each C ∈ C, AC is a finite set of SFSASes of Q[u,y] such that:
(i) either AC is empty, which means that S(u) is empty for each u∈C;
(ii) or AC ={[∅,{ }]}, which implies that cs(u) is infinite for each u∈C;
(iii) or each A = [T, P>] ∈ AC satisfies mvar(T) = y, mvar(P) = y and for
each u∈C we have:
– the associated regular systems of AC specialize well at u,
– for each A∈ AC, ZR(A(u)) is not empty,
– S(u) = ·∪A∈ACZR(A(u)).
If we further require in (iii) that C is a connected semi-algebraic set, then we call (C,(AC, C ∈ C)) a RCTD with connected cells.
Remark 10.1. The RCTD we proposed in paper [34] is essentially a RCTD with con- nected cells as defined above. Algorithm 38 and 40 presented hereafter also compute
1
The “R” in the term RCTD emphasizes the fact this tool focuses on the real solutions of the input parametric polynomial system.
a RCTD with connected cells. Nevertheless, a default specification not requiring con- nectivity provides more flexibility for some applications such as real root classification and projection.
Remark 10.2. By Theorem 8.1 of Section 8.2 in Chapter 8, the condition (iii)in the definition of RCTD implies that above each connected component of the cell C, the solutions of S w.r.t. y are finitely many continuous functions of the parameters u. Moreover, the graphs of these functions are disjoint above each connected component of C.
In the rest of this section, we provide an algorithm for computing a RCTD. It relies on an operation called CAD for decomposing real constructible sets into connected and cylindrically arranged cells of Rn. The operation CAD can be easily described via the three subroutines MPD, MakeCylindrical and MakeSemiAlgebraic presented in Chapter 7. The correctness of the operations CAD follow immediately from those of its three subroutines.
Calling sequence. CAD(C)
Input. C := {C1, . . . , Ce} is a set of pairwise disjoint constructible sets of Cn given
by polynomials in Q[x] such thatCn =∪e i=1Ci.
Output. A CAD2 E of Rn such that for each element C of C, the set C ∩Rn is a union of some cells inE.
Step (1). For 1 ≤ i ≤ e, apply operation MPD to the family of regular systems representing Ci, so as to obtain another family Ri of regular systems representingCi
and whose zero sets are pairwise disjoint. Step (2). Let R := ∪e
i=1Ri. Call algorithm MakeCylindrical(R, n), to compute a
cylindrical decompositionD of Cn such that the zero set of each regular system inR is a union of some cells in D.
Step (3). Call algorithm MakeSemiAlgebraic to compute a CAD E of Rn such that, for each element D of D, the set D∩Rn is a union of some cells in E.
Next we describe algorithms for computing CTD of a semi-algebraic system.
We start by describing an algorithm for computing CTD of a basic semi- algebraic system, see Algorithm 38.
We then present a general algorithm for computing CTD of anarbitrary semi- algebraic system, see Algorithm 40.
Algorithm 38: RCTD(S)
Input: A basic semi-algebraic system S:= [F, P>, H6=] of Q[u,y]. Output: ACTD of S.
begin
1
let (C,(TC, C ∈ C)) be a WDSCTD of the associated constructible set ofS
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D :=CAD(C);E :={ }
3
for each C∈ C, for each D∈ D such that D⊆C, let TD =TC
4 for D∈ D do 5 if TD ={ } then 6 E :=D;AE :={ }; E :=E ∪ {E} 7 else if TD ={∅} then 8 E :=D;AE :={[∅,{ }]}; E :=E ∪ {E} 9 else 10
let s be a sample point of D
11
E :=D;E :=E ∪ {E}; AE :={ }
12
let Py be the set of polynomials in P such that mvar(p)∈y 13
if (P \Py)> is true after evaluating at s then
14 for T ∈ TD do 15 if RealRootIsolate(T, Py)6= [ ] then 16 AE :=AE ∪ {[T, Py>]} 17 return (E,(AE, E∈ E)) 18 end 19 Algorithm 39: RegularizeInequalities(S)
Input: A semi-algebraic system S= [F, N≥, P>, H6=] of a polynomial ring R Output: A set L of triples [T′, P′, H′], where T′ is a set of regular chains ofR,
P′ and H′ are set of polynomials in R, such that: each polynomial in P′∪H′ is
regular w.r.t. every regular chain in T′; ∪[T′,P′,H′]∈L∪T′∈T′ Z(T′, H′) is the associated constructible set of S; and
ZR(S) = ·∪[T′,P′,H′]∈L∪T′∈T′ WR(T′)∩ZR(P>′, H6=′ ). begin 1 T:=Triangularize(F); 2 L :={ [T,∅]}; 3 for p∈N do 4 for [T′, P′]∈ L do 5 L:=L ∪ { [∪T∈T′Intersect(p, T), P′] }; 6 L:=L ∪ { [T′, P′∪ {p}]}; 7 L :={ [T′, P′∪P, H∪P ∪P′]|[T′, P′]∈ L }; 8 L :={ [∪T′∈T′RegularOnly(T′, H′), P′, H′]|[T′, P′, H′]∈ L }; 9 return L 10 end 11
Algorithm 40: RCTD(S)
Input: A semi-algebraic system S= [F, N≥, P>, H6=] of a polynomial ring R Output: A comprehensive triangular decomposition ofS
begin 1 L :=RegularizeInequalities(S);L0 :={ }; L1 :={ }; 2 for [T′, P′, H′]∈ L do 3 R:={ }; 4 for T′ ∈T′ do 5 if y ⊆mvar(T′) then R :=R ∪ {[T′, H′]}; 6 else L1 :=L1∪ {[T′, H′]}; 7 L0 :=L0∪ {[R, P′]}; 8 L0 :={[DSPCTD(R), P′]|[R, P′]∈ L0}; L0 :=∪[R,P′]∈L0 ∪rs∈R[rs, P′]; 9
let cs1 be the projection of the constructible set L1 on Cd;
10 C :=∅; 11 for [rs, P′]∈ L 0 do 12 C :=Difference(Du(rs), cs 1);if C 6=∅ then C :=C ∪ {C}; 13 C :=SMPD(C); 14 for C∈ C do 15
if C is not empty then
16
let AC be the set of [rs, P′]∈ L0 with C ⊆Du(rs);
17 AC :={[Ty, P′]|[[T, H′], P′]∈ L0} 18 C :=cs1;C :=C ∪ {C}; AC :={[∅,{ }]}; 19 C :=Cd\ ∪C∈CC; C :=C ∪ {C}; AC :={ }; 20 D :=CAD(C); 21
for each C∈ C, for each D∈ D such that D⊆C, let AD =AC;
22 E :={ }; 23 for D∈ D do 24 if AD ={ } or AD :={[∅,{ }]} then 25 E :=D;AE :={ }; E :=E ∪ {E}; 26 else 27
let s be a sample point of D;
28 E :=D;E :=E ∪ {E}; AE :={ }; 29 for [T′, P′]∈ A D do 30 let P′
y be the set of polynomials in P′ such that mvar(p)∈y; 31
if (P′\P′
y)(s)> is true and RealRootIsolate(T′(s), Py′(s))6= [ ] 32 then AE :=AE ∪ {[T′, Py′]}; 33 return (E,(AE, E∈ E)) 34 end 35
To prove the correctness of the algorithms, we first establish the following lemma. Lemma 10.1. Let k[u,y] be a polynomial ring. Let T be a regular chain of k[u,y] such that mvar(T) = y. Let p be a polynomial of k[u,y]. Let u ∈ Kd such that T specializes well at u and such that p(u, y)6= 0 holds for any (u, y) ∈ W(T). Then p is regular modulo sat(T).
Proof. Since T specializes well at u and p(u, y) 6= 0 for any (u, y) ∈ W(T), the polynomialp(u,y) is invertible (and thus regular) modulohT(u,y)i. Thus, the regular system [T, p] specializes well atu. Proposition 6.9 implies that res(p, T)(u)6= 0 holds. Therefore res(p, T)6= 0 holds too and p is regular modulo sat(T).
Proposition 10.1. Algorithm 38 terminates and satisfies its specification.
Proof. By the specification of WDSCTD and Lemma 10.1, it is easy to deduce that each element of AE is an SFSAS. Then the termination and the correctness of the
algorithm follow directly from the specifications of its subroutines and the definition of a RCTD.
Proposition 10.2. Algorithm 39 terminates and satisfies its specification. Proof. We have the following observations
Algorithm Triangularize compute a set of regular chains T such that V(F) = ∪T∈TW(T).
For each p∈N, line 6 and 7 consider respectively the casep= 0 and p6= 0. For a regular chainT ∈T
′and a polynomialp∈N, algorithmIntersectcompute
regular chainsT1, . . . , Ts such thatV(p)∩W(T)⊆ ∪si=1W(Ti)⊆V(p)∩W(T),
moreover W(Ti)⊆W(T)⊆V(F).
For a regular chain T ∈T
′ and a set of polynomials H′, algorithm RegularOnly
computes regular chains T1, . . . , Tt such that Z(T, H′) ⊆ ∪ti=1Z(Ti, H′) ⊆
Z(F, H′).
From the above arguments, we can easily deduce the conclusion.
Proposition 10.3. Algorithm 40 terminates and satisfies its specification.
Proof. Firstly, similar to the proof of algorithm 38, each element of AE is an SFSAS.
Secondly, algorithm 39 decomposes the input system as disjoint systems. Then the termination and correctness of the algorithm follows easily from the specifications of DSPCTD and other subroutines.
2
That is, finitely many constructible sets ofRn