Generalized Taylor formulae, computations in real closed valued fields and quantifier elimination Mari-Emi Alonso

∗

Henri Lombardi

†

2002

Abstract We use generalized Taylor formulae in order to give some simple constructions in the real closure of an ordered valued field. We deduce a new, simple quantifier elimination algorithm for real closed valued fields and some theorems about constructible subsets of real valuative affine space.

Key words: Valued fields, Real closed fields, Generalized Taylor formulae, Quantifier elimination, Constructive mathematics MSC 2000: 14P10, 12J10, 12L05, 12Y05, 03F65, 03C10

Introduction In this work, we consider the real closure of an ordered valued field and search for simple computations giving a constructive content to this real closure. We don’t try to give sophisticated algorithms which would allow better complexity. We consider an ordered valued field (K, V, P) with V its valuation ring and P its positive cone. Recall that this means that the following properties hold V + V ⊆ V, V × V ⊆ V, ∃x ∈ K \ V, ∀x, y ∈ K (xy = 1 ⇒ x ∈ V ∨ y ∈ V), P × P ⊆ P, P + P ⊆ P, ∃x ∈ K \ P, ∀x, y ∈ K (x + y = 0 ⇒ x ∈ P ∨ y ∈ P), ∀x, y ∈ K [(x + y ∈ V, x ∈ P, y ∈ P) ⇒ x ∈ V]. For a, b in K, we write a ≤ b if and only if b − a is in P. We shall use freely in the sequel some well known features of ordered valued fields: Q ⊆ V, elements of V bounded from below by some positive rational are units in V, and the non-units in V are the infinitesimal elements of K. ∗

Universitad Complutense, Madrid, Espa˜ na. Partially supported by: PB95/0563-A. M− [email protected] Laboratoire de Math´ematiques, UMR CNRS 6623. Univ. de Franche-Comt´e, France. [email protected] †

1

2 Let S be a subring of V such that K is the fraction field of S. We assume that S is an explicit ordered ring and that divisibility inside V is testable for two arbitrary elements of S. These are our minimal assumptions of computability. If we want more assumptions in certain cases we shall make them explicit. We denote the real closure of (K, P) by (Krc , Prc ), and we write Vrc for the convex hull of V inside Krc ; then Vrc is the unique order-compatible valuation ring extending V. We call (Krc , Vrc , Prc ) the real closure of (K, V, P). In sections 1 and 2 our general purpose is to discuss computational problems in (Krc , Vrc , Prc ) under our computability assumptions on (K, V, P). Each computational problem we shall consider has as input a finite family (ci )i=1,...,n of parameters in the ring S. We call them the coefficients of our computational problem. Our algorithms with the previous minimal computability assumptions work uniformly. This means that some computations are made that give polynomials in Z[C1 , . . . , Cn ], and that all our tests are of the two following types: Is P (c1 , . . . , cn ) ≥ 0?

Does Q(c1 , . . . , cn ) divide P (c1 , . . . , cn ) in V?

We are not interested in how the answers to these tests are found. We may imagine these answers given either by some oracles or by some algorithms. Let us state precisely some other notations. We shall denote the unit group of V by UV , and MV = V \ UV will be the maximal ideal. We shall denote the residue field V/MV of (K, V) by K, and the value group, K× /UV , by ΓK . We use freely the value group’s usual additive notation as well as its usual group-ordering (also denoted by ≤). Recall that ΓKrc is the divisible hull Γdh K of ΓK . For x ∈ K we write v(x) or vK (x) the valuation of x in ΓK ∪ {+∞}. So v(0) = +∞, v(xy) = v(x) + v(y), (x ≥ 0, y ≥ 0) ⇒ v(x + y) = min(v(x), v(y)), and ∀x ∈ K ((v(x) ≥ 0 ⇔ x ∈ V) ∧ (v(x) > 0 ⇔ x ∈ MV )). √ ac ac We write Kac = Krc [ −1] √ and we denote2by V2 the natural valuation ring of K extending rc rc V : for a, b ∈ K , v(a + b −1) = (1/2)v(a + b ). In fact elements of Γdh K ∪ {+∞} are always defined through elements of S in the following form. We say that the valuation of some element x belonging to Kac is well determined if we know integers m and n, elements c1 , ..., cn in S, and two elements F and G of Z[C1 , ..., Cn ], such that, setting f = F (c1 , . . . , cn ) (f 6= 0) and g = G(c1 , . . . , cn ), there exists a unit u in Vac with: f xm = ug (in particular, v(0) is well determined). We read the previous formula as: m vKac (x) = vK (g) − vK (f ), or more simply as: m v(x) = v(g) − v(f ). For x, y ∈ Kac , we shall use the notation x  y for v(x) ≤ v(y) (i.e., y = x = 0 ∨ (x 6= 0 ∧ y/x ∈ Vac )).

3 Example. Let us explain the computations that are necessary to compare 3v(x1 ) + 2v(x2 ) to 7v(x3 ) when the valuations are given by m3 m2 1 f 1 xm 1 = u1 g1 , f2 x2 = u2 g2 , f3 x3 = u3 g3

(g1 , g2 , g3 6= 0).

We consider the LCM m = m1 n1 = m2 n2 = m3 n3 of m1 , m2 , m3 . We have n3 n3 n3 m n2 n2 n2 m n1 n1 f1n1 xm 1 = u1 g1 , f2 x2 = u2 g2 , f3 x3 = u3 g3 .

So 3v(x1 ) + 2v(x2 ) ≤ 7v(x3 ) iff g13n1 g22n2 f37n3  f13n1 f22n2 g37n3 . The reader can easily verify that computations we shall run in the value group are always meaningful under our computability assumptions on the ring S. In the same way, elements of the residue field will in general be defined from elements of V. So computations inside the residue field are given by computations inside S. We now give an outline of the paper. In section 1 we give some basic tools used in the rest of the paper. First we recall the Newton Polygon Algorithm and the Generalized Tschirnhaus Transformation. Then we insist on Generalized Taylor formulae, which are formulae giving P (x) on a Thom interval as a sum of terms all having the same sign. This feature allows us to give a good description for v(P (x)) with the crucial Theorem 1.3.6. This allows us to give a nice description for “constructible” subsets of the real line in the context of real closed valued fields (cf. Theorem 1.4.4). In section 2 we settle three basic computational problems in the real closure of an ordered valued field. We solve the first problem by a simple trick (subsection 2.3). The consequence is that when we know how to compute in a given ordered valued field, we know how to compute in its real closure. This can be seen as a not too difficult extension of basic algorithms in real closed fields. Solving the second problem is possible by using our first algorithm, but we prefer to develop another algorithm, similar to the Cohen-H¨ormander algorithm for ordered fields. We get in this way nice uniform results describing precisely some generalizations of the complete tableau of signs in the real closed case (Theorems 2.4.5 and 2.5.2). In section 3 we give parametrized versions of previous algorithms (Theorems 3.1.1 and 3.1.3), and we apply these results to quantifier elimination in real closed valued fields. We consider the first order theory of real closed valued fields based on the language of ordered fields (0, 1, +, −, ×, =, ≤) to which we add the predicate x  y. So, all constants and variables represent elements in K (this corresponds to our previously explained computability assumptions). We get the following theorem. Theorem 3.2.1 Let Φ(a, x) be a quantifier free formula in the first order theory of real closed valued fields. We view the ai ’s as parameters and the xj ’s as variables. Then one can give a quantifier free formula Ψ(a) such that the two formulae ∃x Φ(a, x) and Ψ(a) are equivalent in the formal theory. (The terms appearing in the formulae Φ and Ψ are Z–polynomials in the parameters, and, in the case of Φ, also in the variables.) We think we have given here a rather simple proof of this fundamental, well known result (see e.g., [2]). We also get the following abstract form of the previous theorem. Theorem 3.3.2 Let us denote the real-valuative spectrum of a commutative ring A by SpervA. Then the canonical mapping from SpervA[X] to SpervA transforms any constructible subset into a constructible subset. In section 4 we apply the parametrized algorithms in order to study constructible subsets (in the meaning of real closed valued fields). First, we get the analogue of the Tarski-Seidenberg principle.

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Theorem 4.1.1 Let (K, V, P) be an ordered valued subfield of a real closed valued field (R, VR , PR ). Let π be the canonical projection from Rn+r onto Rn . Let S ⊆ Rn+r be any (≤, )–constructible set defined over (K, V, P). Assume that the sign test and the divisibility test are explicit inside the ring generated by the coefficients of the polynomials that appear in the definition of S. Then a description of the projection π(S) ⊆ Rn can be computed in a uniform way by an algorithm that uses only rational computations, sign tests and divisibility tests. In particular, the complexity of a description of π(S) is explicitly bounded in terms of the complexity of a description of S. Finally we construct a kind of stratification for (≤, )–constructible sets, that we call stratification a` la Cohen Hormander because it is a further development of the same notion for semialgebraic sets (cf. [1] chapter 9), and we finish the paper with the following cell-decomposition theorem (for a precise definition of Q-semilinear functions see definition 2.4.4). Theorem 4.2.5 (Cell decomposition theorem) Let (K, V, P) be an ordered valued subfield of a real closed valued field (R, VR , PR ). Let g1 , . . . , gs be nonzero polynomials in K[x1 , . . . , xn ]. Consider a linear change of variables together with a family (fi,j )i=1,...,n;j=1,...,`i that give a stratification for (g1 , . . . , gs ). Assume that this stratification is constructed `a la Cohen-H¨ormander. Consider any k-dimensional stratum Cε corresponding to this stratification. Then there is a Nash isomorphism h : (R+ )k −→ Cε , (t1 , . . . , tk ) 7−→ h(t1 , . . . , tk ) with the following property. If S is any (≤, )–constructible subset described from g1 , . . . , gs , then S ∩ Cε is a finite union of cells h(Li ), where each Li can be defined as ( ) ^ ^ (t1 , . . . , tk ) ∈ (R+ )k : a` (τ ) = α` ∧ bm (τ ) > βm , `

m

where τ = (τ1 , . . . , τk ) = (v(t1 ), . . . , v(tk ), the a` ’s and bm ’s are Z-linear forms w.r.t. τ , and α` , βm ∈ Γdh K. Moreover, each τi is a Q-semilinear function in some v(Fj (x1 , . . . , xn ))’s (with Fj explicitly computable in K[x1 , . . . , xn ]).

1 1.1

Basic material The Newton Polygon

Here we recall the well known Newton Polygon algorithm. A multiset is a set with (nonnegative) multiplicities, or equivalently a list defined up to permutation. E.g., the roots of a polynomial P (X) repeated according to multiplicities form a multiset in the algebraic closure of the base field. We shall use the notation [x1 , . . . , xd ] for the multiset corresponding to the list (x1 , . . . , xd ). The cardinality of a multiset is the length of a corresponding list, i.e., the sum of multiplicities occurring in the multiset. P i The Newton polygon of a polynomial P (X) = i=0,...,d pi X ∈ K[X] (where pd 6= 0) is obtained from the list of pairs in N × (ΓK ∪ {+∞}) ((0, v(p0 )), (1, v(p1 )), . . . , (d, v(pd ))). The Newton polygon is “the bottom convex hull” of this list. It can be formally defined as the extracted list ((0, v(p0 )), . . . , (d, v(pd ))) verifying: two pairs (i, v(pi )) and (j, v(pj )) are two consecutive vertices of the Newton polygon iff:

1.2 Generalized Tschirnhaus transformation

5

if 0 ≤ k < i then (v(pj ) − v(pi ))/(j − i) > (v(pi ) − v(pk )/(i − k)) if i < k < j then (v(pk ) − v(pi ))/(k − i) ≥ (v(pj ) − v(pi ))/(j − i) if j < k ≤ d then (v(pk ) − v(pj ))/(k − j) > (v(pj ) − v(pi ))/(j − i) It is easily shown that if (i, v(pi )) and (j, v(pj )) are two consecutive vertices in the Newton polygon of the polynomial P , then the zeroes of P in Kac whose valuation in Γdh K equals (v(pi ) − v(pj ))/(j − i) form a multiset with cardinality j − i. Computational problem 0 (Multiset of valuations of roots of polynomials) Input: Let P ∈ K[X] be a polynomial over a valued field (K, V). Output: The multiset [v(x1 ), . . . , v(xn )] where [x1 , . . . , xn ] is the multiset of roots of P in Kac . Newton Polygon algorithm The number n∞ of roots equal to 0 (i.e., with infinite valuation) is read on P . Let P0 := P/X n∞ . Compute the Newton polygon of P0 , compute the slopes and output the answer.

1.2

Generalized Tschirnhaus transformation

We recall here the well known (generalized) Tschirnhaus transformation, which we will use freely in our computations. Let K be a field, (Pj )j=1,...,m be a family of monic polynomials in K[X], and Pj (X) = (X − xj,1 ) × · · · × (X − xj,dj ) their decompositions in Kac . Let Q(Y1 , . . . , Ym ) be a polynomial in K[Y1 , . . . , Ym ]. Then the polynomial TQ (Z) = (Z − Q(x1,1 , . . . , xm,1 )) × · · · × (Z − Q(x1,d1 , . . . , xm,dm )) is the characteristic polynomial of AQ where AQ is the matrix of the multiplication by Q(y1 , . . . , ym ) inside the d-dimensional K-algebra K[y1 , . . . , ym ] := K[Y1 , . . . , Ym ]/ hP1 (Y1 ), . . . , Pm (Ym )i (d = d1 · · · dm , and yi is the class of Yi modulo hP1 , . . . , Pm i). Now let R ∈ K[Y1 , . . . , Ym ] with R(x) 6= 0 for all m-tuples x = (x1,r1 , . . . , xm,rm ). So AR is an invertible matrix. Let F = Q/R, then the polynomial TF (Z) = (Z − F (x1,1 , . . . , xm,1 )) × · · · × (Z − F (x1,d1 , . . . , xm,dm )) is the characteristic polynomial of AQ (AR )−1 .

1.3

Generalized Taylor Formulas

Using the usual Taylor formula for computing valuations in Γdh K. For P ∈ K[X] we denote P [k] = P (k) /k!, where P (k) is the k-th derivative of P . Let t = x−a, and assume deg(P ) = d, the usual Taylor formula at the point a is P (x) = P (a) + P [1] (a)t + P [2] (a)t2 + · · · + P [d−1] (a)td−1 + P [d] td . Now assume that P [d] > 0. Let a0 be the greatest real root of the product P P [1] · · · P [d−1] . If a ≥ a0 we see that all P [k] (a) are ≥ 0 and we get the following expression for the valuation v(P (x)) when x > a v(P (x)) = min(ν0 , ν1 + τ, ν2 + 2τ, . . . , νd + dτ )

6

1 BASIC MATERIAL

where τ = v(t) and νj = v(P [j] (a)) (some νj ’s may be infinite). So, w.r.t. the variable τ the valuation of P (x) in Γdh K is piecewise linear and increasing. Note that τ decreases from +∞ to −∞ when t increases from 0 to +∞. In the following paragraphs, we see that generalized Taylor formulae allow us to give a similar description of the valuation v(P (x)) when x is inside a Thom interval. What are generalized Taylor formulae? A fundamental example of algebraic evidence for a sign is given by generalized Taylor formulae, which make explicit some consequences of Thom’s lemma in terms of algebraic identities. Thom’s lemma implies that the set of points where a real polynomial and its successive derivatives have fixed signs is an interval. An easy proof, by induction on the degree of the polynomial, is based on the mean value theorem. We can translate this geometric fact under the form of algebraic identities called Generalized Taylor Formulas (GTF for short). Let us see an example where deg(P ) ≤ 4. Example 1.3.1 Consider the general polynomial of degree 4 P (X) = c0 X 4 + c1 X 3 + c2 X 2 + c3 X + c4 , consider the following system of sign conditions for the polynomial P and its successive derivatives with respect to the variable X: H(X) : P (X) > 0, P [1] (X) < 0, P [2] (X) < 0, P [3] (X) < 0, P [4] > 0. Consider also the system of sign conditions obtained by relaxing all the inequalities, except one of them, e.g., the last one: H 0 (X) : P (X) ≥ 0, P [1] (X) ≤ 0, P [2] (X) ≤ 0, P [3] (X) ≤ 0, P [4] > 0. Thom’s lemma implies that: [ H 0 (a), H 0 (b), a < x < b ] =⇒ H(x). Put e1 = x − a, e2 = b − x. Consider the following algebraic identity in Z[c0 , . . . , c4 , a, b, x] P (x) = P (b) − e2 P [1] (a) − (2e1 e2 + e2 2 ) P [2] (a) −(3e1 2 e2 + 3e1 e2 2 + e2 3 ) P [3] (b) +(8e1 3 e2 + 12e1 2 e2 2 + 12e1 e2 3 + 3e2 4 ) P [4] . This gives clearly an evidence that, when P ∈ K[X] where K is an ordered field, [ H 0 (a), H 0 (b), a < x < b ] =⇒ P (x) > 0. One can find more information about mixed and generalized Taylor formulae in [6, 10, 11]. The important thing is that for any fixed degree, and any combination of signs for P and its derivatives (which are assumed to be fixed on the interval), there exists a corresponding GTF. We state a general result giving the existence of GTF’s. Proposition 1.3.2 (see [10]) Let P be a polynomial of degree d in K[X] and a, b, x three variables. Let e1 = x − a, e2 = b − x. Let ε = (1 , . . . , d ) be any sequence in {−1, +1}. Let 0 = 1. Then there exists an algebraic identity P (x) = P (a0 ) +

d−1 X k=1

k Hk,ε (e1 , e2 )P [k] (ak ) + d Hd,ε (e1 , e2 )P [d]

1.3 Generalized Taylor Formulas

7

where each polynomial Hk,ε is homogeneous of degree k with nonnegative integer coefficients, ak = a if k k+1 = 1, and ak = b if k k+1 = −1. Moreover, if 1 = 1, then e1 divides all the Hk,ε ’s, and the coefficient of ek1 in Hk,ε is nonzero. In a similar way if 1 = −1, then e2 divides all the Hk,ε ’s, and the coefficient of ek2 in Hk,ε is nonzero. Remark 1.3.3 Let P be a polynomial of degree d in K[X] and let a < b ∈ Krc be such that P [k] (a)P [k] (b) ≥ 0 for k = 0, . . . , d. This gives a system of signs (σ0 , σ1 , . . . , σd ) (σi = ±1) (σk is the sign of P [k] (x) on the open interval ]a, b[ ). Let i = σ0 σi , ε = (1 , . . . , d ). Then the corresponding GTF gives an algebraic certificate for the fact that sign(P (x)) = σ0 when a < x < b. We now give four GTF’s in degree 3, those beginning by P (a) + e1 P [1] · · ·. Each formula is given also with e1 in factor in the second part. P (x) = P (a) + e1 P [1] (a) + e21 P [2] (a) + e31 P [3]
= P (a) + e1 P [1] (a) + e1 P [2] (a) + e21 P [3] = P (a) + e1 P [1] (a) + e21 P [2] (b) − (2e31 + e21 e2 ) P [3]
= P (a) + e1 P [1] (a) + e1 P [2] (b) − (2e21 + e1 e2 ) P [3] = P (a) + e1 P [1] (b) − (e21 + 2e1 e2 ) P [2] (a) − (2e31 + 6e21 e2 + 3e1 e22 ) P
[3] = P (a) + e1 P [1] (b) − (e1 + 2e2 ) P [2] (a) − (2e21 + 6e1 e2 + 3e22 ) P [3] = P (a) + e1 P [1] (b) − (e21 + 2e1 e2 ) P [2] (b) + (e31 + 3e21 e2 + 3e1 e22 ) P
[3] = P (a) + e1 P [1] (b) − (e1 + 2e2 ) P [2] (b) + (e21 + 3e1 e2 + 3e22 ) P [3] . There are also four other GTF’s beginning by P (b) − e2 .P [1] · · ·. They can be obtained from the first ones by swapping a and b, and replacing e1 and e2 by −e2 and −e1 P (x) = P (b) − e2 P [1] (b) + e22 P [2] (b) − e32 P [3]
= P (b) − e2 P [1] (b) − e2 P [2] (b) + e22 P [3] = P (b) − e2 P [1] (b) + e22 P [2] (a) + (2e32 + e22 e1 ) P [3]
= P (b) − e2 P [1] (b) − e2 P [2] (a) − (2e22 + e2 e1 ) P [3] = P (b) − e2 P [1] (a) − (e22 + 2e2 e1 ) P [2] (b) + (2e32 + 6e22 e1 + 3e2 e21 ) P
[3] = P (b) − e2 P [1] (a) + (e2 + 2e1 ) P [2] (b) − (2e22 + 6e2 e1 + 3e21 ) P [3] = P (b) − e2 P [1] (a) − (e22 + 2e2 e1 ) P [2] (a) + (e32 − 3e22 e1 + 3e2 e21 ) P
[3] = P (b) − e2 P [1] (a) + (e2 + 2e1 ) P [2] (a) + (e22 + 3e2 e1 + 3e21 ) P [3] . Using generalized Taylor formulae for computing the variations of the valuation v(P (x)). Now let us see in the case of an ordered valued field how these formulae can be used in order to describe the variations of v(P (x)) when x is on the real line Krc . Example 1.3.4 Let a, b ∈ Krc and assume that the signs of the derivatives of a polynomial P of degree 4 are the same in a and b, as in Example 1.3.1. If x ∈ [a, b] let x = a + t1 (b − a),

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e = b − a, e1 = t1 e, e2 = t2 e (so t2 = 1 − t1 ), δ = v(e), τ1 = v(t1 ), τ2 = v(t2 ), ν0 = v(P (b)), ν1 = v(P [1] (a)), ν2 = v(P [2] (a)), ν3 = v(P [3] (b)), ν4 = v(P [4] ). We rewrite the GTF as P (x) = P (b) − e t2 P [1] (a) − e2 (2t1 t2 + t2 2 ) P [2] (a) −e3 (3t1 2 t2 + 3t1 t2 2 + t2 3 ) P [3] (b) +e4 (8t1 3 t2 + 12t1 2 t2 2 + 12t1 t2 3 + 3t2 4 ) P [4] . In the above GTF, since all terms of the sum are ≥ 0, the valuation of the sum is the minimum of valuations of the terms, so we get: (1) If t1 and t2 are units, then τ1 = τ2 = 0 and v(P (x)) is constant equal to v(P (x)) = min(ν0 , ν1 + δ, ν2 + 2δ, ν3 + 3δ, ν4 + 4δ). (2) If t1 is infinitely close to 0, then τ1 > 0 (decreasing as t1 increases), τ2 = 0, and v(P (x)) is a priori increasing “piecewise linearly w.r.t. τ1 ”, but in our case constant v(P (x)) = min(ν0 , ν1 + δ, ν2 + 2δ, ν3 + 3δ, ν4 + 4δ). (3) If t1 is infinitely close to 1, then τ1 = 0, τ2 > 0 (increasing as t1 increases), and v(P (x)) is increasing “piecewise linearly w.r.t. τ2 ” v(P (x)) = min(ν0 , ν1 + δ + τ2 , ν2 + 2δ + τ2 , ν3 + 3δ + τ2 , ν4 + 4δ + τ2 ). In fact here we see that this formula is true in the three cases and that only two slopes (w.r.t. the variable τ2 ) can appear since v(P (x)) = min(ν0 , min(ν1 + δ, ν2 + 2δ, ν3 + 3δ, ν4 + 4δ) + τ2 ). Example 1.3.5 In a similar way let us see what is given by the second GTF in degree 3
P (x) = P (a) + e1 P [1] (a) + e21 P [2] (b) − 2e31 + e21 e2 P [3] . We assume P (a) ≥ 0, P [1] (a) ≥ 0, P [2] (b) ≥ 0, P [3] < 0, x = a + t1 (b − a), e = b − a, e1 = t1 e, e2 = t2 e (t2 = 1 − t1 ), δ = v(e), τ1 = v(t1 ), τ2 = v(t2 ), ν0 = v(P (a)), ν1 = v(P [1] (a)), ν2 = v(P [2] (b)), ν3 = v(P [3] ), and we get
P (x) = P (a) + e t1 P [1] (a) + e2 t21 P [2] (b) − e3 t31 + t21 t2 P [3] . (1) If t1 and t2 are units, then τ1 = τ2 = 0 and v(P (x)) is constant equal to v(P (x)) = min(ν0 , ν1 + δ, ν2 + 2δ, ν3 + 3δ). (2) If t1 is infinitely close to 1, then τ1 = 0, τ2 > 0 (increasing as t1 increases), and v(P (x)) is increasing “piecewise linearly w.r.t. τ2 ”, but in our case constant v(P (x)) = min(ν0 , ν1 + δ, ν2 + 2δ, ν3 + 3δ). (3) If t1 is infinitely close to 0, then τ1 > 0 (decreasing as t1 increases), τ2 = 0, and v(P (x)) is increasing “piecewise linearly w.r.t. τ1 ”, v(P (x)) = min(ν0 , ν1 + δ + τ1 , ν2 + 2δ + 2τ1 , ν3 + 3δ + 2τ1 ). In fact here we see that this formula is true in the three cases and that only three slopes (w.r.t. the variable τ1 ) can appear since v(P (x)) = min(ν0 , (ν1 + δ) + τ1 , min(ν2 + 2δ, ν3 + 3δ) + 2τ1 ).

1.4 Constructible subsets of the real line

9

What we have seen on our two Examples 1.3.4 and 1.3.5 is a general result, that we immediately get as a corollary of Proposition 1.3.2. Theorem 1.3.6 Let P be a polynomial of degree d in K[X] and a < b ∈ Krc such that P [k] (a)P [k] (b) ≥ 0 for k = 0, . . . , d. Let (σ0 , σ1 , . . . , σd ) be the signs of P, P [1] , . . . , P [d] in the interval ]a, b[ (σi = ±1). Let i = σ0 σi , ε = (1 , . . . , d ). Let us consider the corresponding GTF as in Proposition 1.3.2, and let us follow the notation there. Let νk = v(P [k] (ak )) for k = 0, . . . , d. Recall that ak = a if k k+1 = 1, and ak = b if k k+1 = −1. Note also that νk may be infinite if k < d. If x ∈ [a, b] let x = a + t1 (b − a), e = b − a, t2 = 1 − t1 , δ = v(e), τ1 = v(t1 ), τ2 = v(t2 ). Then for x ∈ [a, b] the valuation v(P (x)) is monotonic w.r.t. t1 and more precisely can be described in the following way. (a)

– If 1 = 1 we can extract from the GTF integers k2 , . . . , kd such that 1 ≤ kj ≤ j and v(P (x)) = min(ν0 , ν1 + δ + τ1 , ν2 + 2δ + k2 τ1 , . . . , νd + dδ + kd τ1 ). – If 1 = −1 we can extract from the GTF integers k2 , . . . , kd such that 1 ≤ kj ≤ j and v(P (x)) = min(ν0 , ν1 + δ + τ2 , ν2 + 2δ + k2 τ2 , . . . , νd + dδ + kd τ2 ).

(b) So in any case the valuation v(P (x)) is – either constant (if v(P (a)) = v(P (b))), ) (if v(P (a)) > v(P (b))), – or increasing piecewise linearly w.r.t. τ1 = v( x−a b−a – or increasing piecewise linearly w.r.t. τ2 = v( b−x ), (if v(P (a)) < v(P (b))). b−a (c) Introducing  τ = τ1 − τ2 = v

t1 1 − t1



we also get: τ1 = max(τ, 0) = τ + , τ2 = max(−τ, 0) = τ − , and the value v(P (x)) is monotone and piecewise linear w.r.t. τ . More precisely, we can extract from the GTF integers k2 , . . . , kd such that 1 ≤ kj ≤ j and v(P (x)) = min(ν0 , ν1 + δ + τ 0 , ν2 + 2δ + k2 τ 0 , . . . , νd + dδ + kd τ 0 ) where τ 0 = max(1 τ, 0).

1.4

Constructible subsets of the real line

We introduce here the notion of (≤, )–constructible sets in the real valuative affine space. This notion corresponds to sets that are definable in the language of ordered valued fields. These sets are analogous to Zariski-constructible sets in algebraic geometry and to semi-algebraic sets in real algebraic geometry. Definition 1.4.1 Let (K, V, P) be an ordered valued field, and consider a finite family (xj )j=1,...,m of elements of Krc . Let us call a valued sign condition (a vsc fort short) for the family any condition of the following type   ^ ^ X sign(xj ) = σj ∧ sign  `j v(xj ) = σ`0 j∈J

`∈L

j∈J, σj 6=0

10

1 BASIC MATERIAL

where J ⊆ {1, . . . , m}, ` ∈ L (L is a finite subset of Z{j : j∈J, σj 6=0} ) and σj , σ`0 ∈ {−1, 0, 1}. Let N be a positive integer. We call an N -complete system of valued sign conditions on the a system of vsc’s that gives all the signs sign(xj ) and all the signs Pfamily (xj )j=1,...,m  {j : 1≤j≤m, xj 6=0} sign . xj 6=0 `j v(xj ) for all ` ∈ {−N, . . . , 0, . . . , N } P  An alternative definition could use sign ` v(x ) even when v(xj ) = ∞ for some j’s. j j∈J, j But there should be no natural way to give a sign to an expression containing ∞ − ∞. Definition 1.4.2 Let (K, V, P) be an ordered valued subfield of a real closed valued field (R, VR , PR ), and consider a finite family (Pj )j=1,...,m of polynomials in K[X1 , . . . , Xn ]. • The subset of Rn made of the x = (x1 , . . . , xn ) such that the Pj (x)’s verify some given system of vsc’s is called a basic (≤, )–constructible set defined over (K, V, P). • A (general) (≤, )–constructible set defined over (K, V, P) is any boolean combination S of basic (≤, )–constructible sets defined over (K, V, P). If (Pj )j=1,...,m is a family of polynomials such that any basic component of S is defined as in the first item, we say that S is described from (Pj )j=1,...,m . • Let S ⊆ Rn be a (≤, )–constructible set. A map f : S → Rp is called a (≤, )–constructible map if its graph is a (≤, )–constructible subset of Rn+p . Let us recall that the order topology and the valued topology are identical in a real closed valued field. Notation 1.4.3 Let (K, V, P) be an ordered valued subfield of a real closed valued field (R, VR , PR ). We shall use the following notations for some convex open (≤, )–constructible subsets of the real line. They are basic (≤, )–constructible sets defined over (Krc , Vrc , Prc ). I+ (a, α) = {x ∈ R : x = a + t, 0 < t, v(t) = α} with a ∈ Krc , α ∈ Γdh K. − I (a, α) = {x ∈ R : x = a − t, 0 < t, v(t) = α} with a ∈ Krc , α ∈ Γdh K. I+ (a, α, β) = {x ∈ R : x = a + t, 0 < t, α < v(t) < β} with a ∈ Krc , α < β in Γdh K ∪ {±∞}. − I (a, α, β) = {x ∈ R : x = a − t, 0 < t, α < v(t) < β} with a ∈ Krc , α < β in Γdh K ∪ {±∞}. + J (a, b, α) = {x ∈ R : x = a + t(b − a), 0 < t, v(t) = α} with a < b ∈ Krc , 0 < α ∈ Γdh K. − J (a, b, α) = {x ∈ R : x = b − t(b − a), 0 < t, v(t) = α} with a < b ∈ Krc , 0 < α ∈ Γdh K. + J (a, b, α, β) = {x ∈ R : x = a + t(b − a), 0 < t, α < v(t) < β} with a < b ∈ Krc , 0 ≤ α < β ∈ Γdh K ∪ {+∞}. − J (a, b, α, β) = {x ∈ R : x = b − t(b − a), 0 < t, α < v(t) < β} with a < b ∈ Krc , 0 ≤ α < β ∈ Γdh K ∪ {+∞}. J(a, b) = {x ∈ R : x = a + t(b − a), 0 < t < 1, v(t) = v(1 − t) = 0} with a < b ∈ Krc . These subsets will be called ( 0 and v(t) > 0. • Except when β = ∞, any ( 0, α < β ∈ Γdh K ∪ {+∞}, c > 0 ∈ K and v(c) = α + β we have the following equivalences α < v(t) < β ⇐⇒ α < min(v(t), v(c/t)) ⇐⇒ α < v(t + c/t) ⇐⇒ α + v(t) < v(t2 + c). • Concerning J(a, b) we have J(a, b) = {x ∈ R : x = a + t(b − a), 0 < t(1 − t), v(t(1 − t)) = 0} . • All J’s could be considered as particular cases of I’s, e.g., J+ (a, b, α, β) = I+ (a, α0 , β 0 ) with α0 = α + v(b − a) and β 0 = β + v(b − a). • We could introduce J(a, b, α) = {x ∈ R : x = a + t(b − a), 0 < t < 1, v(t/(1 − t)) = α} with a < b ∈ Krc , α ∈ Γdh K, J(a, b, α, β) = {x ∈ R : x = a + t(b − a), 0 < t < 1, α < v(t/(1 − t)) < β} with a < b ∈ Krc , α < β in Γdh K ∪ {±∞}. We should have J+ (a, b, α) = J(a, b, α), J− (a, b, α) = J(a, b, −α), J+ (a, b, α, β) = J(a, b, α, β), J− (a, b, α, β) = J(a, b, −β, −α) and J(a, b) = J(a, b, 0). An easy corollary of Theorem 1.3.6 is the following description of (≤, )–constructible subsets of the real line. Theorem 1.4.4 Let (K, V, P) be an ordered valued subfield of a real closed valued field (R, VR , PR ). Any (≤, )–constructible set of R defined over (K, V, P) is a finite disjoint union of points in Krc and of ( 2λ1 + λ4 we have two vertices on the polygonal graph of v(P3 ) at the points with coordinates τ1,1 = β1 = (λ1 − λ4 )/3, τ1,2 = β2 = λ0 − λ1 ,

v(P3 (x)) = λ1 + β1 = λ4 + 4β1 , v(P3 (x)) = λ0 = λ1 + β2 .

All these vertices give a finite number of valuations for τ1 : α1 < · · · < αn . Let α0 = 0, αn+1 = ∞. On each J+ (a, b, αi , αi+1 ) (0 ≤ i ≤ n) and on each J+ (a, b, αi ) (1 ≤ i ≤ n), we know that each v(Pj (x)) (1 ≤ j ≤ 3) is a fixed “affine function” of τ1 . So, the same is true for any linear combination `1 v(P1 (x)) + `2 v(P2 (x)) + `3 v(P3 (x)), and we can compute the valuation τ1 for which such an expression changes sign. So the intersection S ∩ J+ (a, b, 0, ∞) is a finite disjoint union of J+ (a, b, α, β) and J+ (a, b, α) subsets. In a similar way S ∩ J(a, b) is either empty or equal to J(a, b), and S ∩ J− (a, b, 0, ∞) is a finite disjoint union of J− (a, b, α, β) and J− (a, b, α) subsets. Finally the intersection of S with the final (resp. initial) open interval is computed in a similar way as a finite union of I+ (resp. I− ) intervals. 2

2 2.1

Computing in the real closure of an ordered valued field Codes ` a la Thom and valuations in the value group

The real closure Krc of an ordered field (K, P) is unique up to unique (K, P)–isomorphism. This fact allows us to give an explicit construction of the real closure Krc (this is “well-known” from Tarski or even from Sturm and Sylvester, for a fully constructive proof see [7]). E.g., it is possible to describe any element x of Krc by a so-called code `a la Thom (see [3, 4]): Definition 2.1.1 A pair (P, σ) where P ∈ K[X] is a monic polynomial of degree d and σ = (σ1 , . . . , σd−1 ) ∈ {1, −1}d−1 codes the root x of P in Krc when one has P (x) = 0

and

σi · P (i) (x) ≥ 0

for i = 1, . . . , d − 1.

The pair (P, σ) is called a code `a la Thom (over K) for x. There are algorithms that use only the algebraic structure of (K, P) and give the codes a` la Thom corresponding to the roots of P in Krc . It is possible to make explicit algebraic computations and sign’s tests for such elements that are coded a` la Thom. See e.g., [3, 4] or Proposition 2.4.2. On the other hand, the Newton polygon algorithm allows us to determine the valuation v(x) for any x in the algebraic closure of K. How can we match these algorithms?

2.2 Three basic computational problems in the real closure of an ordered valued field

2.2

13

Three basic computational problems in the real closure of an ordered valued field

Consider an ordered valued field (K, V, P). Since its real closure (with valuation) is determined up to unique (K, V, P)-isomorphism, the following computational problems makes sense: Computational Problem 1 Let (K, V, P) be an ordered valued field. Input: A code `a la Thom (P, σ) over K for an element x of Krc . Output: The valuation v(x) of x in Γdh K ∪ {+∞}. More precisely, compute some a ∈ K and a positive integer n such that n × v(x) = v(a). Remark 2.2.1 Assume that the leading coefficient of P ∈ V[X] is a unit. The real zeroes of P are in Vrc . Let us denote by x the residue in Krc of the zero x and by P the residue in K[X] of the polynomial P . Then it is clear that (P , σ) is a code `a la Thom over K for x since the rc residual field Krc can be identified with the real closure K of K. More generally, we can ask for algorithms solving general existential problems. Computational Problem 2 Let (K, V, P) be an ordered valued field, and consider a finite family of polynomials, (Fj )j=1,...,m in K[X]. Let (xh )h=1,...,p be the ordered family of the zeroes of the (Fj )’s in Krc . Recall that the number p and all the signs sign(Fj (x)), for x equal to some xh or inside some corresponding open interval, can be determined by computations in the ordered field (K, P). Input: The family (Fj )j=1,...,m . Output: All the valuations v(Fj (xh )) (h = 1, . . . , p) and v(xh+1 − xh ) (h = 1, . . . , p − 1) in Γdh K ∪ {+∞}. Computational Problem 3 Let (K, V, P) be an ordered valued field. Input: A finite family (Fj )j=1,...,m in K[X]. A finite family (`k )k=1,...,r of elements of Zm . Output: All occurring systems of valued sign conditions of the following type for the family (Fj (x))j=1,...,m when x ∈ Krc : 





(sign(Fj (x)))j=1,...,m , sign 

 X j∈{1,...,m}, Fj (x)6=0



`k,j v(Fj (x))

. k=1,...,r

Remark 2.2.2 Assume that the family is stable under derivation. From Theorem 1.3.6 (see e.g., the proof of Theorem 1.4.4) it is clear that Computational Problem 3 can be solved by using the solution of Computational Problem 2. In fact we can describe in a finite way all occurring lists ( (sign(Fj (x)))j=1,...,m , (v(Fj (x)))j=1,...,m ) when x ∈ Krc : for x on any ( 0 (else replace P by P (−X)) and that P is monic. Let (xi )i=1,...,d be the roots of P in Kac . Using the Newton Polygon algorithm, we compute the multiset [ v(xi ) | i = 1, . . . , n ]. So we can express the set of valuations v(xi ) as (v(cj )/nj )j=1,...,r for some r-tuple (cj , nj )j=1,...,r with cj > 0 in K, nj ∈ N and v(cj )/nj < v(cj+1 )/nj+1 for j = 1, . . . , r − 1. Consider the LCM n of denominators nj and “replace each xi by zi = xni ”: i.e., compute Q n/n Q(X) = i (X − zi ) and compute a code a` la Thom (Q, σ 0 ) for z = xn . Let bj = cj j . Then v(bj ) = (n/nj )v(cj ) for j = 1, . . . , r and v(b1 ) < · · · < v(br ). So we have also b1 > · · · > br > 0. By rational computations in (K, P) we can settle one of the three following inequalities in Krc bj ≥ z ≥ bj+1

z ≥ b1 , with some j ∈ {1, . . . , r − 1}, br ≥ z > 0.

In the first case we conclude that v(z) = v(b1 ). In the last case v(z) = v(br ). In the remaining case we know that v(bj ) ≤ v(z) ≤ v(bj+1 )

so v(z) = v(bj ) or v(z) = v(bj+1 ).

We have to find the exact valuation. Consider c ∈ P verifying      bj+1 bj ,v if j > 1 0 < v(c) ≤ min v bj−1 bj   b2 0 < v(c) = v if j = 1 b1 (if j > 1, c can be chosen as bj /bj−1 or bj+1 /bj ). Next consider the linear fractional change of variable y y 7−→ ϕ(y) = 1 + cy 2 We have — If v(y) ≥ 0 then v(ϕ(y)) = v(y). — If v(y) ≤ −v(c) then, letting y 0 = 1/y we get v(y 0 ) ≥ v(c) > 0, ϕ(y) =

y0 c + y02

and

v(ϕ(y)) = v(y 0 ) − v(c) ≥ 0.

So the monic polynomial R(Y ) =

Y i

  zi Y −ϕ bj

has coefficients in V. Moreover v(z/bj ) ≥ 0, so ϕ(z/bj ) is a unit iff v(bj ) = v(z) since v(ϕ(z/bj )) = v(z/bj ).

2.4 Solving the second problem

15

We can compute a code `a la Thom (R, σ 00 ) for ϕ(z/bj ). This gives a code `a la Thom (R, σ 00 ) rc for ϕ(z/bj ) (i.e., ϕ(z/bj ) considered as an element of K ). Finally we test whether this code is verified by 0 (which is a root of R). In case of negative answer then v(z) = v(bj ). Otherwise v(z) = v(bj+1 ). Remarks 2.3.1 1) In a more explicit view, we should ask for computing two nonnegative elements a and b of K and an integer n such that a ≤ |x|n ≤ b and v(a) = v(b). 2) Clearly algorithm RCVF1 allows us to run sure computations inside (Krc , Vrc , Prc ) when we know how to compute inside (K, V, P).

2.4

Solving the second problem

First we recall the Cohen-H¨ormander algorithm for ordered fields (see e.g., [1] chapter 1). Definition 2.4.1 Let (K, P) be an ordered field and (Fj ) a finite family of univariate polynomials in K[X]. A complete tableau of signs for the family (Fj ) is the following discrete data T: • The ordered list (xk )k=1,...,r of all the roots of all the Fj ’s in Krc . • The signs (∈ {−1, 0, +1}) of all the Fj ’s at all the xk ’s. • The signs of all the Fj ’s in each interval ] − ∞, x1 [ , ]xk , xk+1 [ (1 ≤ k ≤ r − 1) and ]xr , +∞[ . We call an xk a point of the tableau T . Similarly an interval ] − ∞, x1 [ or ]xk , xk+1 [ or ]xr , +∞[ is called an interval of the tableau T . In this tableau xk is merely a name for the corresponding root, it may be coded by the number k or in another way. Proposition 2.4.2 (Cohen-H¨ormander’s algorithm for computing the complete tableau of signs for a finite family of univariate polynomials) Let (K, P) be an ordered subfield of a real closed field (R, PR ). Let L = (F1 , . . . , Fk ) be a list of polynomials in K[Y ]. Let L0 be the family of polynomials generated by the elements of L and by the operations P 7→ P 0 and (P, Q) 7→ Rem(P, Q) for deg(P ) ≥ deg(Q) ≥ 1. Then L0 is finite and one can compute the complete tableau of signs for L0 in terms of the following data: • the degree of each polynomial in the family L0 , • the diagrams of operations P 7→ P 0 and (P, Q) 7→ Rem(P, Q), • the signs of constants ∈ L0 . L et us remark that in this algorithm the zero polynomial can appear in L0 as a remainder Rem(P, Q) where deg(P ) ≥ deg(Q) ≥ 1. The degree of the zero polynomial is −1. The list L0 is finite: one makes systematically the operation “derivation of every previously obtained polynomial” and “remainders of all previously obtained couple of polynomials”, and one gets a finite family at the end since degrees are decreasing. Let us number the polynomials in L0 with an order compatible with the order on the degrees. Let L0m be the subfamily of L0 made of polynomials numbered from 1 to m. This

16

2 COMPUTING IN THE REAL CLOSURE OF AN ORDERED VALUED FIELD

family is obviously stable under the operations “derivation” and “remainder by a division” which decrease strictly the degrees. Denote lastly by Tm the corresponding complete tableau of signs. We are going to prove, by induction on m, that the complete tableau of signs of the polynomials in the family L0m can be obtained by using only the authorized informations. As long as polynomials are of degree 0, this is clear. Suppose it is true up to m. Let P be the polynomial of number m + 1 in L0 . On each interval of Tm , the polynomial P is strictly monotonic. Every point a of Tm is either +∞, or −∞, or a root of a certain polynomial Q with number ≤ m, and in this case, if R = Rem(P, Q), we have P (a) = R(a). The sign of P (a) is hence known in every case from the authorized informations. This allows us to know on which open intervals of Tm the polynomial P has a root in R. Let x be such a root of P on one of these open intervals I = ]a, b[ . If Q is a polynomial of number ≤ m in P , its sign on the interval I is known. This means we know its sign at the point x, and on intervals ]a, x[ and ]x, b[. With respect to P , its signs on ]a, x[ and on ]x, b[ are also known since P is strictly monotonic on the interval. The complete tableau of signs for L0m+1 is thus known from the authorized informations and the complete tableau of signs for L0m . 2 In this algorithm we remark that each zero of the tableau is obtained with a Thom’s encoding. An extension of previous algorithm will solve Problem 2. First we give a valued version for the complete tableau of signs. Definition 2.4.3 Let (K, V, P) be an ordered valued field and (Fj )j∈J a finite family of univariate polynomials in K[X]. A complete tableau of vsc’s for the family (Fj ) is the following data T : • The ordered list (xk )k=1,...,r of all the roots of all the Fj ’s in Krc . • The complete tableau of signs for the family (Fj )j∈J . • All the valuations v(xk+1 − xk ) (k = 1, . . . , r − 1). • All the valuations v(Fj (xk )) (j ∈ J, k = 1, . . . , r). Algorithm RCVF2 solving Problem 2. A first possibility is to use algorithm RCVF1. We think that it is interesting to indicate another possibility which goes in the same spirit as the Cohen-H¨ormander algorithm for ordered fields. This gives us also simple proofs for theorems in sections 3 and 4. Call (Pj ) the list L0 in Proposition 2.4.2. Call (xm,k )k=1,...,rm the ordered list of all roots of L0m = (Pj )j=1,...,m . We replace in the proof of Proposition 2.4.2 the complete tableau of signs Tm of L0 by Sm = Tm ∪ Vm where Vm collects the valuations v(Pj (xm,k )) (j ∈ {1, . . . , m}, k ∈ {1, . . . , rm }) and v(xm,k+1 − xm,k ) (k ∈ {1, . . . , rm − 1}.) Suppose we have done the job up to m. Let P = Pm+1 be the polynomial of index m + 1 in L0 . The tableau Tm+1 is computed as in Proposition 2.4.2. It remains to compute missing informations in Vm+1 . At every root a = xm,k of a polynomial Q = P` with index ` ≤ m, if R = Rem(P, Q), we have P (a) = R(a) and R is in L0m , so the valuation v(P (a)) is known from Vm . Let x = a + t1 (b − a) be a root of P on an open interval I = ]xm,k , xm,k+1 [ = ]a, b[ of Tm . In order to compute all the v(Pj (x))j=1,...,m it is sufficient to compute v(t1 ) = τ1 and v(t2 ) = τ2 (t2 = 1 − t1 ): Theorem 1.3.6 says us how to get the valuations v(Pj (x))j=1,...,m from Vm , τ1 and τ2 .

2.4 Solving the second problem

17

In order to compute τ1 = v(t1 ) we use a GTF that expresses P (x) = P (a + t1 (b − a)) = 0 as ! d X P (a) + t1 · j · ej · Gj,ε (t1 , t2 ) · P [j] (aj ) (aj = a or b) j=1

where e = b − a, t1 · Gj,ε (t1 , t2 ) = Hj,ε (t1 , t2 ) and sign(j P [j] (aj )) = sign(−P (a)) (1 ≤ j ≤ d). Moreover, the valuations v(P (a) = ν, v(P [j] (aj )) = νj and δ = v(b − a) are known. From the properties of Hj,ε , we know that Gj,ε (t1 , t2 ) is a unit if τ1 = 0, so its valuation in Γdh K ∪ {+∞} depends only on τ1 . So we get v(P (a)) = ν = min(ν1 + δ + τ1 , ν2 + 2δ + k2 τ1 , . . . , νd + dδ + kd τ1 ) (τ1 ≥ 0, and some νk ’s may be infinite). The right hand side is an increasing piecewise linear function of τ1 so we have a unique and explicit solution τ1 . With µi = νi + iδ we precisely get   ν − µd ν − µ2 . ,..., τ1 = max ν − µ1 , k2 kd Finally τ2 is computed analogously and we can fill up Vm+1 . Remark also that if x is on the last interval ]xm,rm , +∞[ = ]a, +∞[ of Tm , we can compute v(x − a) in a similar way by using the usual Taylor formula. Definition 2.4.4 In an additive divisible ordered group G we consider terms built from variables αj by Q-linear combinations and by using the operations min and max. We call such a term a Q-semilinear term. The function defined by such a term is called a Q-semilinear function of the αj ’s. We get the following theorem, similar to Proposition 2.4.2. Theorem 2.4.5 (An algorithm `a la Cohen-H¨ormander for computing the complete tableau of vsc’s for a finite family of univariate polynomials) Let (K, V, P) be an ordered valued subfield of a real closed valued field (R, VR , PR ). Let L = (F1 , . . . , Fk ) be a list of polynomials in K[Y ]. Let L0 be the (finite) family of polynomials generated by the elements of L and by the operations P 7→ P 0 and (P, Q) 7→ Rem(P, Q) for deg(P ) ≥ deg(Q) ≥ 1. Call (cj ) the list of constants ∈ L0 . Then one can compute the complete tableau of vsc’s for L0 in terms of the following data: • the degree of each polynomial in the family, • the diagrams of operations P 7→ P 0 and (P, Q) 7→ Rem(P, Q) in L0 , • the signs sign(cj ), • the valuations v(cj ). Moreover, all the valuations v(xk+1 − xk ) and all the valuations v(Pj (xk )) are given as fixed Q-semilinear functions of the v(cj )’s: each such Q-semilinear function is a fixed Q-semilinear term (in the “variables” v(cj )’s) that depends only on the complete tableau of signs of L0 . T his theorem is an extension of Proposition 2.4.2. The proof is similar. In fact we get all results by a close inspection of Algorithm RCVF2. 2

18

2 COMPUTING IN THE REAL CLOSURE OF AN ORDERED VALUED FIELD

2.5

Solving the third problem

Algorithm RCVF3 solving Problem 3. We run Algorithm RCVF2 and we apply Theorem 1.3.6: see Remark 2.2.2. Definition 2.5.1 Let (Fj )j∈J be a finite family of univariate polynomials in K[X] (where (K, V, P) is an ordered valued field). We assume the family to be stable under derivation. Let M be a positive integer. An M -complete tableau of vsc’s for the family (Fj ) is the following discrete data T : • The ordered list (xk )k=1,...,r of all the roots of all the Fj ’s in Krc . • For each k = 1, . . . , r, the M -complete system of vsc’s (see Definition 1.4.1) for the family (Fj (xk ))j∈J . • For each k = 1, . . . , r − 1 – The M -complete system of vsc’s for the family (Fj (x))j∈J for x ∈ J(xk , xk+1 ). – A partition of J+ (xk , xk+1 , 0, ∞) as a finite union of 2nk + 1 ( 0 ∧ v(ϕ(c )) = v(ϕ(d )) ∧ v(ϕ(e )) > v(ϕ(f )) i j k k ` ` i j k ` where conjunctions are finite and all elements are in B. Searching the canonical image of a basic constructible subset S of SpervA[X] (defined by elements ai , bj , ck , dk , e` , f` in A[X]) inside SpervA, is the same thing that analyzing the conditions on the coefficients of the polynomials ai , bj , ck , dk , e` , f` allowing the existence of an x where the defining conditions of S are verified. So Theorem 3.1.3 gives the answer. 2 Another consequence of Theorem 3.1.3 is a relativized version of Theorem 3.3.2. This generalization is obtained by giving some constraints on the ring homomorphism ϕ from A to a real closed valued field K. We give e.g., a subring B of A, an ideal M of B, a multiplicative monoid S in A and a semi ring P in A (P + P ⊆ P, P × P ⊆ P ). We want to allow only homomorphisms φ (from A or A[X] to a real closed valued field) verifying that φ(B) is in the valuation ring, φ(M ) is in the maximal ideal, elements of φ(S) are nonzero and elements of φ(P ) are nonnegative. If we write C the constraints (B, M, S, P ) and if we write Sperv(A, C) the part of SpervA satisfying the constraints, we get: the canonical mapping from Sperv(A[X], C) to Sperv(A, C) transforms any (≤, )–constructible subset in a (≤, )–constructible subset. In [9] the relativized version is settled with one constraint B.

4 4.1

Constructible subsets in the real valuative affine space Tarski-Seidenberg-Chevalley

We now give a geometric form for Theorems 3.1.3 and 3.2.2. Theorem 4.1.1 Let (K, V, P) be an ordered valued subfield of a real closed valued field (R, VR , PR ). Let π the canonical projection from Rn+r onto Rn . Let S ⊆ Rn+r be any (≤, )– constructible set defined over (K, V, P). Assume that the sign test and the divisibility test

22

4 CONSTRUCTIBLE SUBSETS IN THE REAL VALUATIVE AFFINE SPACE

are explicit inside the ring generated by the coefficients of the polynomials that appear in the definition of S. Then a description of the projection π(S) ⊆ Rn can be computed in a uniform way by an algorithm that uses only rational computations, sign tests and divisibility tests. In particular, the complexity of a description of π(S) is explicitly bounded in terms of the complexity of a description of S. Here rational computations mean computations in the ring generated by the coefficients of the polynomials occurring in the description of S. A description of S is a quantifier free formula in disjunctive normal form describing S. The complexity of such a description of S can be defined as a 5-tuple (n, d, k, `, m) where n is the number of variables, d is the maximum of the degrees, k is the number of polynomials, ` is the number of ∨ and m is the bound for the numbers of ∧ inside a disjunct. Corollary 4.1.2 Let (K, V, P) be an ordered valued subfield of a real closed valued field (R, VR , PR ). Let S ⊆ Rn be a (≤, )–constructible set and let f : S → Rp be a (≤, )–constructible map. • The interior and the adherence of S inside Rn for the order topology are (≤, )–constructible sets. • f (S) ⊆ Rp is a (≤, )–constructible set. • Let T be a (≤, )–constructible set containing f (S) and let g : T → Rq be a (≤, )–constructible map. Then g ◦ f is a (≤, )–constructible map. • Let T 0 ⊆ Rp be a (≤, )–constructible set. Then f −1 (T 0 ) ⊆ Rn is a (≤, )–constructible set.

4.2

Stratifications and applications

We think that the results of this section could allow to get most of the results obtained by Frank Mausz in his Doctoral dissertation [8] with a different approach. Lojaziewicz stratification ` a la Cohen-H¨ ormander We recall here a result about stratifying families ([1] chapter 9). Definition and notation 4.2.1 Consider a general monic polynomial of degree d as a point of Rd . Let σ = (σ1 , . . . , σd ) ∈ {−1, +1}d . Let ( !) d ^ Uσ = P ∈ Rd : ∃x ∈ R P (x) = 0 ∧ sign(P (i) (x)) = σi . i=1

It is easily seen that Uσ is a connected open semialgebraic subset of Rd (see e.g., [5]) and that ( !) d ^ Uσ = P ∈ Rd : ∃x ∈ R P (x) = 0 ∧ sign(P (i) (x)) ∈ {σi , 0} . i=1

For P ∈ Uσ we call ρσ (P ) the zero which is coded `a la Thom by (P, σ). Then P 7→ ρσ (P ) is Nash on Uσ and admits a continuous semialgebraic extension on Uσ , that we note also by ρσ . Such a function will be called a Thom’s root function, or simply a root function. More generally, if ϕ : Rk−1 → Rd is a polynomial function, we can consider ρσ ◦ ϕ as defined over ϕ−1 (Uσ ). We also call such a function a root function. This function is Nash

4.2 Stratifications and applications

23

over ϕ−1 (Uσ ). If f (x1 , . . . , xk ) = ϕ(x1 , . . . , xk−1 )(xk ) is the corresponding monic polynomial in k variables, we denote ρσ ◦ ϕ by ρσ (f ). Finally if a polynomial g ∈ K[x1 , . . . , xk ] = K[x1 , . . . , xk−1 ][xk ] has a leading coefficient w.r.t. xk which is a nonzero element c of K, we say that g is quasi monic in xk , and we let ρσ (g) = ρσ (g/c). For more details about root functions see [5]. Theorem 4.2.2 ([1] chap. 9) Let (K, P) be an ordered subfield of a real closed field (R, PR ). Let g1 , . . . , gs be nonzero polynomials in K[x1 , . . . , xn ]. After a suitable linear change of variables there exists a family of polynomials (fi,j )i=1,...,n;j=1,...,`i with the following properties (we will continue denoting the new variables by xi ). (1) First we have – (g1 , . . . , gs ) ⊆ (fn,j )j=1,...,`n – Each fk,j is a nonzero polynomial in K[x1 , . . . , xk ] which is quasimonic in xk . – For each index k the family (fk,j )j=1,...,`k is stable under derivation w.r.t. xk (excluding the zero derivative). (2) Let us denote Ik = {(i, j) : i = 1, . . . , k; j = 1, . . . , `i }. Call Ck the family of nonempty semialgebraic subsets of Rk that can be defined as some     ^ k Cε = (ξ1 , . . . , ξk ) ∈ R ; sign(fi,j (ξ1 , . . . , ξk )) = i,j 6= ∅   (i,j)∈Ik

(where ε = (i,j )(i,j)∈Ik is any family in {−1, 0, +1}). It is clear that the Cε ’s in Ck give a partition of Rk . We have (a) The canonical projection πk (Cε ) of any element Cε ∈ Ck on Rk−1 is an element of Ck−1 : it is obtained as Cε0 where ε0 is the restriction of the family ε to Ik−1 . (b) The adherence Cε of Cε (recall we assume Cε 6= ∅) is a union of elements of Ck , it is obtained by relaxing strict inequalities in the definition of Cε . (c) If in the definition of Cε ∈ Ck there is one equality fk,i (ξ1 , . . . , ξk ) = 0 then Cε is the graph of a root function ρσ (fk,j ) (here fk,j is seen as a polynomial in xk , it is (`) equal to fk,i or to some fk,i and σ is extracted from ε) which is Nash over πk (Cε ). Moreover, ρσ (fk,i ) is defined over πk (Cε ) and the graph of this root function is Cε . (d) Call πn,k the canonical projection Rn → Rk . Let E be a k dimensional semialgebraic subset of Rn defined from the polynomials g1 , . . . , gs . Then for any Cε ∈ Cn which is contained in E, πn,k maps homeomorphically Cε on its image. Definition 4.2.3 Such a change of variables together with such a family (fi,j ) will be called a stratification for (g1 , . . . , gs ) and for any semialgebraic subset of Rn defined from this family. The family (fi,j )i=1,...,n;j=1,...,`i will be called a stratifying family for the initial family (g1 , . . . , gs ). The semialgebraic subsets Cε are called the strata of the stratification.

24

4 CONSTRUCTIBLE SUBSETS IN THE REAL VALUATIVE AFFINE SPACE

We shall precisely consider the following way of constructing a stratifying family, `a la CohenH¨ormander (it is the one suggested in [1].) First we make a linear change of variables in order to make g1 , . . . , gs quasi monic in the new variable xn . We add all the derivatives of each gi w.r.t. xn . This gives us the family (fn,j )j=1,...,`n . We apply Cohen-Hormander’s algorithm to this family and we call h1 , . . . , h` the “constants” given by this algorithm (these constants are polynomials in (x1 , . . . , xn−1 )). We make a new linear change of variables on (x1 , . . . , xn−1 ) in order to make h1 , . . . , h` quasi monic in the new variable xn−1 . We make the same linear change of variables inside (fn,j )j=1,...,`n : this family remains quasimonic in xn and stable under derivation w.r.t. xn , and h1 , . . . , h` remain the “constants” given by the Cohen-Hormander’s algorithm when applied to this family. We add all the derivatives of each hi w.r.t. xn−1 . This gives us the family (fn−1,j )j=1,...,`n−1 . And so on. With this kind of stratifying family, we can apply recursively Theorem 3.1.3. So we get a precise description of the variation of the valuations v(fk,j (x1 , . . . , xk )) when (x1 , . . . , xk ) ∈ Cε for any k and any Cε ∈ Ck . Let us see an example. Example 4.2.4 Assume n = 3. Consider a cell C ∈ C3 . Assume that C 00 = π3,1 (C) is an interval ]a, b[ , that C 0 = π3,2 (C) is the graph of a root function h1 = ρσ (f2,1 ) defined on [a, b], and that C is the part of C 0 × R between two root functions h2 = ρσ0 (f3,1 ) and h3 = ρσ00 (f3,2 ), so C = {(x, y, z) : a < x < b, y = h1 (x), h2 (x, y) < z < h3 (x, y)} = {(x, y, z) : a < x < b, y = h1 (x), h02 (x) < z < h03 (x)} . We consider for (x, y, z) ∈ C, the parameters t = (x − a)/(b − x), τ = v(t), t0 = (z − h02 (x))/(h03 (x) − z) and τ 0 = v(t0 ). We get: • The map h : (t, t0 ) 7→ (x, y, z) ∈ C is a Nash isomorphism from (R+ )2 onto C. • For any fk,j in the stratifying family v(fk,j (x, y, z)) = ϕk,j (τ, τ 0 ) is a Q-semilinear function of τ, τ 0 (here we use recursively Theorem 3.1.3). • So, if we look at C ∩ S where S is any (≤, )–constructible subset described from the fk,j ’s, we find that C ∩ S is a finite union of sets h(Li ) where each Li is defined as ) ( ^ ^ bm (τ, τ 0 ) > βm (t, t0 ) ∈ (R+ )2 : a` (τ, τ 0 ) = α` ∧ `

m

where a` ’s and bm ’s are Z-linear forms and α` , βm ∈ Γdh K. • Now we should like to have some rational expression of τ and τ 0 that uses only polynomials in (x, y, z). This is possible in the following way, as in Remark 3.1.2. Consider that the formal variables are X, Y, Z and that x, y, z are three parameters. Add to the list gi the three polynomials X − x, Y − y, Z − z and reconstruct the stratification, using the information that (x, y, z) is in the semialgebraic set C. You get that τ and τ 0 are fixed Q-semilinear functions in the v(cj )’s and in some v(Fj (x, y, z))’s: the cj ’s are the old constants, and the Fj (x, y, z) are the new “constants” that are constructed by the algorithm (Fj (x, y, z) ∈ K[x, y, z]). The following “cell decomposition theorem” is merely the generalization of what we have seen on this example. It is obtained by applying Theorem 1.3.6 to a stratification a` la CohenH¨ormander. The last assertion is obtained as in Remark 3.1.2.

REFERENCES

25

Theorem 4.2.5 (Cell decomposition theorem) Let (K, V, P) be an ordered valued subfield of a real closed valued field (R, VR , PR ). Let g1 , . . . , gs be nonzero polynomials in K[x1 , . . . , xn ]. Consider a linear change of variables together with a family (fi,j )i=1,...,n;j=1,...,`i that give a stratification for (g1 , . . . , gs ). Assume that this stratification is constructed `a la Cohen-H¨ormander, as explained above (after Definition 4.2.3). Consider any k-dimensional stratum Cε corresponding to this stratification (see Theorem 4.2.2). Then there is a Nash isomorphism h : (R+ )k −→ Cε , (t1 , . . . , tk ) 7−→ h(t1 , . . . , tk ) with the following property. If S is any (≤, )–constructible subset described from g1 , . . . , gs , then S ∩ Cε is a finite union of cells h(Li ), where each Li can be defined as ) ( ^ ^ + k (t1 , . . . , tk ) ∈ (R ) : a` (τ ) = α` ∧ bm (τ ) > βm `

m

where τ = (τ1 , . . . , τk ) = (v(t1 ), . . . , v(tk ), the a` ’s and bm ’s are Z-linear forms w.r.t. τ , and α` , βm ∈ Γdh K. Moreover, each τi is a Q-semilinear function in some v(Fj (x1 , . . . , xn ))’s (with Fj ’s explicitly computable elements of K[x1 , . . . , xn ]).

References [1] Bochnak J., Coste M., Roy M.-F. G´eom´etrie alg´ebrique r´eelle. Springer-Verlag (1987). English version Real Algebraic Geometry. Springer-Verlag (1998) 4, 15, 22, 23, 24 [2] Cherlin, Dickmann M. A., Real closed rings II. Model Theory. Ann. of Pure and Applied Logic 25, (1993) 213–231. 3, 20 [3] Cohen A., Cuypers H., Sterk H. (eds) Some Tapas of Computer Algebra. Springer Verlag (1999). 12 [4] Coste M., Roy M.-F. Thom’s Lemma, the coding of real algebraic numbers and the computation of the topology of semi-algebraic sets. J. of Symbolic Computation 5 (1988), 121-129. 12 [5] Gonz´alez-Vega L., Lombardi H., Mah´e L. Virtual roots of real polynomials. J. of Pure and Applied Algebra 124, (1998) 147–166. 22, 23 [6] Lombardi H. Une borne sur les degr´es pour le Th´eor`eme des z´eros r´eel effectif. in: Real Algebraic Geometry. Lecture Notes in Math. no 1524. Eds.: Coste M., Mah´e L., Roy M.-F.. Springer-Verlag, (1992), pp. 323–345. 6 [7] Lombardi H., Roy M.-F. Constructive elementary theory of ordered fields. in Effective Methods in Algebraic Geometry. Eds.: Mora T., Traverso C.. Birkh¨auser. Basel. 1991. Progress in Math. no 94. pp. 249–262. 12 [8] Mausz F. Definierbare Mengen u ¨ber bewerteten reel abgeschlossenen K¨orpen, Doctoral Dissertation, Univ. K¨oln, 1995. 22 [9] De la Puente M.J. Specializations and a local homeomorphism theorem for real Riemann surfaces of rings. Pacific J. of Math. 176 (2), (1996) 427–442. 21

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CONTENTS

[10] Warou H. An algorithm and bounds for the real effective Nullstellensatz in one variable. Progress in Math. no 143, Birkh¨auser. Basel. 1996. pp. 373–387. 6 [11] Warou H. Formules de Taylor G´en´eralis´ees et applications. Preprint Universit´e de Niamey (1999). 6

Contents Introduction

1

1 Basic material 1.1 The Newton Polygon . . . . . . . . . . . 1.2 Generalized Tschirnhaus transformation 1.3 Generalized Taylor Formulas . . . . . . . 1.4 Constructible subsets of the real line . .

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4 4 5 5 9

2 Computing in the real closure of an ordered valued field 2.1 Codes `a la Thom and valuations in the value group . . . . . . . . . . . . . . . . 2.2 Three basic computational problems in the real closure of an ordered valued field 2.3 Solving the first problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Solving the second problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Solving the third problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

12 12 13 14 15 18

3 Quantifier elimination algorithms 3.1 Parametrized computations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Quantifier elimination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 An abstract form of quantifier elimination . . . . . . . . . . . . . . . . . . . . .

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4 Constructible subsets in the real valuative affine space 21 4.1 Tarski-Seidenberg-Chevalley . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 4.2 Stratifications and applications . . . . . . . . . . . . . . . . . . . . . . . . . . . 22