We construct such a homeomorphism by choosing a small deformation of the coefficients of the original equations

VARIETIES DEFINED OVER NUMBER FIELDS

ADAM PARUSI ´NSKI AND GUILLAUME ROND

Abstract. We show that every affine or projective algebraic variety defined over the field of real or complex numbers is homeomorphic to a variety defined over the field of algebraic numbers. We construct such a homeomorphism by choosing a small deformation of the coefficients of the original equations. This method is based on the properties of Zariski equisingular families of varieties.

Moreover we construct an algorithm, that, given a system of equations defining a varietyV, produces a system of equations with coefficients inQof a variety homeomorphic toV.

1. Introduction

In computational algebraic geometry one is interested in computing the solution set of a given system of polynomial equations, or at least in computing various algebraic or geometric invariants of such a solution set. Here by computing we mean producing an algorithm that gives the desired result when it is implemented in an appropriate computer system.

Such an algorithm should be stable with respect to a small perturbation of the coefficients. But in general the shape of the solution set may change drastically un- der a small perturbation of the coefficients. This difficulty is particularly apparent if one has to deal with polynomial equations whose coefficients are neither rational nor algebraic numbers. The main goal of this paper is to show that, when we are interested in the topology of such a solution set, one can always assume that the polynomial equations have algebraic number coefficients precisely by choosing in an effective way a particular perturbation of its coefficients.

Theorem 1. Let V ⊂Kⁿ (resp. V ⊂Pⁿ_K) be an affine (resp. projective) algebraic set, where K = R or C. Then there exist an affine (resp. projective) algebraic set W ⊂ Kⁿ (resp. W ⊂ Pⁿ_K) and a homeomorphism h : Kⁿ −→ Kⁿ (resp.

h:Pⁿ_K−→Pⁿ_K) such that:

(i) the homeomorphism mapsV ontoW,

(ii) W is defined by polynomial equations with coefficients inQ∩K.

Remark 2. In fact we prove a more precise and stronger result: see Theorem 10 for a precise statement.

In the proof of Theorem 10 we show thatW can obtained by a small deformation of the coefficients of the equations definingV. Let denote these coefficients bygi,α. The deformation is denoted byt7→gi,α(t) withgi,α(0) =gi,α. This deformation is constructed in such a way that every polynomial relation with rational coefficients satisfied by theg_i,αis satisfied by theg_i,α(t) for everyt. So we can prove that the

1

deformation is equisingular in the sense of Zariski since this equisingularity condi- tion can be encoded in term of the vanishing and the non-vanishing of polynomial relations on thegi,α(t). In particular this deformation is topologically trivial.

The first part of this paper is devoted to the proof of an approximation result similar to the one given in [Ro]. In the next part we recall the notion of Zariski equisingularity. The proof of Theorem 10 is given in the following part. Finally we provide an algorithm that, given the equations definingV, computes the equations definingW. This algorithm is based on the proof of Theorem 10.

NotationFor a vector of indeterminates x= (x1, . . . , xn), xⁱ denotes the vector of indeterminates (x1, . . . , xi). To avoid confusion, variables will be denoted by normal letterst and elements ofKwill be denoted by bold letterst.

For a fieldKthe ring of algebraic power series is denoted byKhxi.

2. An approximation result

Let Ω⊂K^rbe open and non-empty and letϕbe an analytic function on Ω. We say that f is a Nash function at a∈ Ω if there exist an open neighborhoodU of ain Ω and a nonzero polynomial P ∈K[t1, . . . , tr, z] such that P(t, ϕ(t)) = 0 for t∈U or, equivalently, if the Taylor series ofϕatais an algebraic power series. An analytic function on Ω is a Nash function if it is a Nash function at every point of Ω. An analytic mappingϕ: Ω→K^N is aNash mapping if each of its components is a Nash function.

We call a Nash function ϕ : Ω→K, Q-algebraic (and we noteϕ ∈QA(Ω)) if (locally) it is algebraic overQ[t], i.e. it satisfies a polynomial relationP(t, ϕ(t)) = 0 for everyt∈Ω with P ∈Q[t1, . . . , tr, z], P 6= 0. It is well known that this means that the Taylor expansion ofϕat a point with coordinates inQ(orQ) is an algebraic power series whose coefficients lie in a common finite field extension ofQ, i.e. this Taylor expansion belongs tokhxiwherek is a subfield ofKwhich is finite overQ (see [RD84] for example).

Theorem 3. Let K =R or C. Let y ∈ K^m\Q

m. Denote by k the field exten- sion of Qgenerated by the coefficients of the components yi of y. The field k is a primitive extension of a transcendental finitely generated field extension of Q: k = Q(t1, . . . ,tr)(z) where the ti ∈ K are algebraically independent over Q and z∈Kis finite of degree doverQ(t₁, . . . ,t_r).

Then there exist new variables t= (t₁, . . . , t_r)andz, an open non-empty neighbor- hood oft,U ⊂K^r, a vector function

y(t, z)∈Q(t)[z]^m and a function z(t)∈QA(U)such that

z=z(t),y=y(t,z),

and for allf(y)∈Q[y]^p, wherey= (y₁, . . . , y_m)is a vector of indeterminates, such that f(y) = 0, we have

f(y(t, z)) = 0.

Moreover, the functionz(t)is algebraic of degreedoverQ(t).

Proof of Theorem 3. Let us denote byk the field extension ofQgenerated by the components of y, so k ⊂ K. Since there are finitely many such coefficients, k is a finitely generated field extension of Q. Let t₁, . . . , t_r be a transcendence basis

of k over Q (with r ≥ 1 because y ∈/ Q

m) and set L := Q(t1, . . . ,tr). By the primitive element theorem, there exists an element z ∈ k finite over L and such thatL(z) =k.

For everyi= 1, . . . , mwe can write yi=

d−1

X

k=0

pi,k(t1, . . . ,tr) q_i,k(t₁, . . . ,t_r)z^k

wheredis the degree ofzoverL,pi,k(t),qi,k(t)∈Q[t] andqi,k(t1, . . . ,tr)6= 0. By multiplying thepi,k by some well-chosen polynomials we may assume that all the qi,k(t) are equal, let us say toq(t).

LetP(t, z)∈Q(t)[z] be the monic polynomial of minimal degree inzsuch that P(t₁, . . . ,t_r,z) = 0.

Sett:= (t1, . . . ,tr)∈K^rand letD⊂K^r be the discriminant locus ofP(t, z) seen as a polynomial in z (i.e. D is the locus of points a ∈ K^r such that a is a pole of one of the coefficients ofP or such thatP(a, z) has at least one multiple root).

SinceP(t1, . . . ,tr, z) has no multiple roots in an algebraic closure ofL, the point tis not inD. Then there existU ⊂K^r\D a simply connected open neighborhood oftand analytic functions

w_i:U −→K, i= 1, . . . , d such that

P(t, z) =

d

Y

i=1

(z−w_i(t)) andw1(t1, . . . ,tr) =z.

Moreover thet7−→wi(t) are algebraic functions overQ[t]. Let us setQK=Qwhen K=RandQK=Q+iQwhenK=C. Thenw₁∈QA(U) and the Taylor series of w₁at a point ofU ∩Q^r_K is an algebraic power series whose coefficients belong to a finite field extension ofQ.

Since the polynomialqis not vanishing attthe function t∈K^r\{q= 0} 7−→ 1

q(t)

is a Nash function whose Taylor series at a point of Q^r_K\{q = 0} is an algebraic power series with rational coefficients.

Letb:= (b₁, . . . , b_r)∈Q^r_K∩ U \{q= 0}. Let us denote byϕ₁(t) the Taylor series of w1. Then letϕ2(t)∈Qhtidenote the Taylor series oft7−→ _q(t)¹ atb. For simplicity we can make a translation and assume thatbis the origin ofK^r.

For everyi= 1, . . . , mlet us define (2.1) yi(t1, . . . , tr, z, v) :=v

d−1

X

k=0

pi,k(t1, . . . , tr)z^k and

y(t, z, v) := (y₁(t, z, v), . . . , y_m(t, z, v)). Letf(y)∈Q[y]^p such thatf(y) = 0. We have that

f(y(t₁, . . . ,t_r,z, 1

q(t))) = 0

or equivalently

f(y(t1, . . . ,tr, ϕ1(t1, . . . ,tr), ϕ2(t1, . . . ,tr))) = 0.

But the function

t7−→f(y(t1, . . . , tr, ϕ1(t1, . . . , tr), ϕ2(t1, . . . , tr)))

is an algebraic function overQ[t] andt1, . . . ,tr are algebraically independent over Q. Thus we have that

f y(t1, . . . , tr, ϕ1(t1, . . . , tr), ϕ2(t1, . . . , tr))

= 0.

Indeed, write f = (f1, . . . , fm) and for each i = 1, . . . , m consider the complex valued function

ϕ(t) =fi y(t1, . . . , tr, ϕ1(t1, . . . , tr), ϕ2(t1, . . . , tr)) .

Note that ϕ∈QA(U) because so is w₁. Let P₁ ∈Q[t₁, . . . , t_r, z] be a polynomial of minimal degree such thatP₁(t, ϕ(t)) = 0. Note thatP₁is irreducible. Write

P₁(t, z) =

degP₁

X

k=0

a_k(t)z^k.

Then a0(t) = 0 since ϕ(t) = 0. Therefore a0(t) = 0 because t1, . . . , tr are algebraically independent overQ. HenceP1=P2z for some polynomialP2, which meansϕ(t) vanishes identically onU.

This proves the theorem by defining

y(t, z) =y(t, z, ϕ₂(t))

andz(t) =w1(t). Indeed the coefficients of the components ofy(t, z, ϕ2(t)) seen as polynomials inz are rational power series by (2.1). Moreover ϕ1(t) belongs to a finite field extension ofQ(t) of degreedsinceP(t, z) is irreducible.

The following lemma will be used in the proof of the main theorem:

Lemma 4. Let ϕ(t) ∈ Chti be a power series algebraic over Q[t] and P(t,Φ) ∈ Q[t,Φ] be a nonzero polynomial of degree d in Φ such that P(t, ϕ(t)) = 0. Let t∈Q^rbe such thattis not in the vanishing locus of the coefficients of P(t,Φ)seen as a polynomial inΦ, and such thattbelongs to the domain of convergence ofϕ(t).

Thenϕ(t)is an algebraic number of degree ≤doverQ.

Proof. The proof is straightforward: just replacetbytin the relationP(t, ϕ(t)) =

0.

3. Zariski equisingularity

Zariski equisingularity of families of singular varieties was introduced by Zariski in [Za71] (originally it was called the algebro-geometric equisingularity). Answering a question of Zariski, Varchenko showed [Va72, Va73, Va75] that Zariski equisingu- lar families are locally topologically trivial. In the papers [Va73, Va75] Varchenko considers the families of local singularities while the paper [Va75] deals with the families of affine or projective algebraic varieties. A new method of proof of topo- logical triviality, giving much stronger statements, was given recently in [PP17].

The case of families of algebraic varieties was considered in sections 5 and 9 of [PP17]. The version presented below follows from the proof of the main theorem, Theorem 3.3, of [PP17], see also Theorems 3.1 and 4.1 of [Va72] in the complex

case and Theorems 6.1 and 6.3 [Va72] in the real case, and Proposition 5.2 and Theorem 9.2 of [PP17] where the algebraic global case is treated.

Theorem 5. [Va72],[PP17] Let V be an open connected neighborhood of t in K^r and letOV denote the ring ofK-analytic functions onV. Lett= (t1, . . . , tr)denote the variables inV and letx= (x1, . . . , xn)be a set of variables inKⁿ. Suppose that fori=k₀, . . . , n, there are given

Fi(t, xⁱ) =x^d_iⁱ+

d_i

X

j=1

a_i−1,j(t, xⁱ⁻¹)x^d_i^i−j∈ O_V[xⁱ],

with di>0, such that

(i) for every i > k₀, the first non identically equal to zero generalized dis- criminant ofF_i(t, xⁱ⁻¹, x_i)equalsF_i−1(t, xⁱ⁻¹)up to a multiplication by a nowhere vanishing function ofO_V.

(ii) the first non identically zero generalized discriminant ofF_k0 is independent of xand does not vanish at t.

Let us set for everyq∈ V,Vq={(q,x)∈ V ×Kⁿ|Fn(q,x) = 0}.

Then for everyq∈ V there is a homeomorphism h_q:{t} ×Kⁿ→ {q} ×Kⁿ such that hq(Vt) =Vq.

Moreover ifFn=G1· · ·Gs then for every j= 1, . . . , s h_q G⁻¹_j (0)∩({t} ×Kⁿ)

=G⁻¹_j (q)∩({0} ×Kⁿ).

Remark 6. The homeomorphismh_qof Theorem 5 is written ash_q(t, x) = (q,Ψ_q(x)) for everyx. In the case where theF_iare homogeneous polynomials with respect to the indeterminatesx, the homeomorphisms Ψ_q satisfy

∀λ∈K^∗,∀x∈Kⁿ Ψq(λx) =λΨq(x)

Hence if we defineP(V_q) ={(q,x)∈ V ×Pⁿ_K |F_n(q,x) = 0}, the homeomorphism hqinduces an homeomorphism betweenP(Vt) andP(Vq).

Remark 7. By construction of [Va72, Va73, Va75] and [PP17], the homeomor- phismshq can be obtained by a local topological trivialization. That is there is a neighborhoodW oftinK^rand a homeomorphism

Φ :W ×Kⁿ→ W ×Kⁿ

such that Φ(q, x) = (q, hq(x)). Moreover, as follows from [PP17], we may require that:

(1) The homeomorphism Φ is subanalytic. In the algebraic case, that is in the case considered in this paper, we replace in the assumptionsOVby the ring ofK-valued Nash functions onV. Then Φ can be chosen semialgebraic.

(2) Φ is arc-wise analytic, see Definition 1.2 of [PP17]. In particular eachhqis arc-analytic. It means that for every real analytic arcγ(s) : (−1,1)→Kⁿ, h_q◦γis real analytic and the same property holds forh⁻¹_q .

4. Generic linear changes of coordinates

Let f ∈ K[x] be a polynomial of degree d and let k be a field extension of Q containing the coefficients off. We denote byf the homogeneous part of degreed off. The polynomial

f(x1, . . . , x_n−1,1)6≡0

otherwisefwould be divisible byx_n−1 which is impossible sincefis a homogeneous polynomial. So there exists

(µ1, . . . , µ_n−1)∈Qⁿ⁻¹

such thatc:=f(µ1, . . . , µn−1,1)6= 0. Then let us denote byϕµ the linear change of coordinates defined by

xi 7−→ xi+µixn fori < n

xn 7−→ xn.

We have that

ϕ_µ(f) =cx^d_n+h

wherehis a homogeneous polynomial of degreedbelonging to the ideal generated byx₁, . . . , x_n−1. So we have that

ϕµ(f) =cx^d_n+

d

X

j=1

aj(xⁿ⁻¹)x^d−j_n for some polynomialsa_j(xⁿ⁻¹)∈k[xⁿ⁻¹].

Remark 8. If g = g1. . . gs is a product of polynomials of k[x], then the linear change of coordinatesϕµ defined above also satisfies

ϕµ(gi) =cix^d_nⁱ+

d

X

j=1

ai,j(xⁿ⁻¹)x^d−j_n for some nonzero constantsci∈k.

Remark 9. For everyf ∈k[x] of degree dwhere k⊂K andµ∈Qⁿ⁻¹, we have that

ϕ−µ◦ϕµ(f) =f.

Let us define the support off =P

α∈Nⁿf_αx^α as

Supp(f) :={α∈Nⁿ |f_α6= 0}.

Let us assume that

ϕµ(f) =X f_α⁰x^α

for some polynomialsf_α⁰ ∈k. Then the coefficientfα has the form fα=Pα(µ1, . . . , µn−1, f_β⁰)

for some polynomialPα∈Z[µ, f_β⁰] depending only on thef_β⁰ with|β|=|α|.

5. Main theorem We can state now our main result:

Theorem 10. LetV ⊂Kⁿ (resp. V ⊂Pⁿ_K) be an affine (resp. projective) algebraic set, where K = R or C. Then there exist an affine (resp. projective) algebraic set W ⊂ Kⁿ (resp. W ⊂ Pⁿ_K) and a homeomorphism h : Kⁿ −→ Kⁿ (resp.

h:Pⁿ_K−→Pⁿ_K) such that:

(i) The homeomorphism mapsV ontoW,

(ii) W is defined by polynomial equations with coefficients inQ∩K,

(iii) The variety W is obtained fromV by a Zariski equisingular deformation.

In particular the homeomorphism hcan be chosen semialgebraic and arc- analytic,

(iv) Let g1,. . . , gs be the generators of the ideal defining V. Let us fix ε >

0. Then W can be chosen so that the ideal defining it is generated by polynomialsg₁⁰,. . . ,g_s⁰ ∈Q[x] such that if we write

g_i= X

α∈Nⁿ

g_i,αx^α andg_i⁰= X

α∈Nⁿ

g⁰_i,αx^α

then for every iandαwe have that:

|gi,α−g⁰_i,α|< ε.

(v) Every polynomial relationship with rational coefficients between thegi,αwill also be satisfied by theg_i,α⁰ .

Remark 11. We will see in the proof the following: letkdenote the field extension ofQgenerated by the coefficients of thegi and assume thatk6=Q(otherwise there is nothing to prove). By the primitive element theorem,kis a simple extension of a purely transcendental extensionLofQ, i.e. Q−→Lis purely transcendental and L−→k is a simple extension of degreed. Then the coefficients of theg_i⁰ belong to a finite extension ofQof degree≤d.

In particular ifQ−→k is purely transcendental thenW is defined overQ. Remark 12. In fact (v) implies that

∀i, α gi,α= 0 =⇒g_i,α⁰ = 0.

Ifεis chosen small enough we may even assume that

∀i, α gi,α= 0⇐⇒g⁰_i,α= 0.

Proof of Theorem 10. Let us consider a finite number of polynomialsg1,. . . , gs ∈ K[x] generating the ideal defining V and let us denote by gi,α their coefficients as written in the theorem. After a linear change of coordinatesϕµ withµ∈Qⁿ⁻¹ as in Section 4 we can assume that

ϕ_µ(g_r) =c_rx^p_n^r+

p_r

X

j=1

b_n−1,r,j(xⁿ⁻¹)x^p_n^r−j = X

β∈Nⁿ

a_n,r,βx^β ∀r= 1, . . . , s.

Moreover by multiplying eachϕµ(gr) by 1/cr we can assume thatcr= 1 for every r. We denote by fn the product of the ϕµ(gr) and by an the vectors of coeffi- cients an,r,β. The entries of an are polynomial functions in the gi,α with rational coefficients, let us say

(5.1) a_n=A_n(g_i,α)

for some An = (An,r,β)r,β ∈ Q(ui,α)^Nn for some integer Nn > 0 and some new indeterminatesui,α.

Letln be the smallest integer such that

∆n,l_n(an)6≡0

where ∆n,l denotes the l-th generalized discriminant offn. In particular we have that

∆n,l(an)≡0 ∀l < ln.

After a linear change of coordinates xⁿ⁻¹ with coefficients in Q we can write

∆n,l_n(an) =en−1fn−1 with fn−1= X

β∈Nⁿ

an−1,βx^β=x^d_n−1ⁿ⁻¹+

dn−1

X

j=1

bn−2,j(xⁿ⁻²)x^d_n−1^n−1−j

for some constantse_n−1, a_n−1,β∈k, withe_n−16= 0, and some polynomialsb_n−2,j ∈ k[xⁿ⁻²]. We denote by a_n−1 the vector of coefficients a_n−1,β. Let l_n−1 be the smallest integer such that

∆_n−1,ln−1(a_n−1)6≡0

where ∆_n−1,l denotes thel-th generalized discriminant off_n−1.

We repeat this construction and define a sequence of polynomials fj(x^j), j = k0, . . . , n−1 for somek0, such that

∆_j+1,lj+1(a_j+1) =e_j



x^d_j^j +

d_j

X

k=1

b_j−1,k(x^j−1)x^d_j^j−k



=e_j



 X

β∈Nⁿ

a_j,βx^β



=e_jf_j is the first non identically zero generalized discriminant of fj+1, whereaj denotes the vectors of coordinatesaj,β in k^Nj, i.e. we have that :

(5.2) ∆j+1,l_j+1(aj+1) =ej



 X

β∈Nⁿ

aj,βx^β



=ejfj

and

(5.3) ∆j+1,l(aj+1)≡0 ∀l < lj+1, until we getf_k0 = 1 for somek₀≥1.

By (5.2) we see that the entries ofa_j ande_j are rational functions in the entries ofaj+1 for every j < n. Thus by (5.1) we see that the entries of theak and theej

are rational functions in thegi,α with rational coefficients, let us say a_k =A_k(g_i,α), e_j=E_j(g_i,α)

for some Ak ∈Q(ui,α)^Nk and Ej ∈Q(ui,α) for everyk and j, where the ui,α are new indeterminates for everyi andα.

By Theorem 3, there exist a new set of indeterminates t = (t1, . . . , tr), an open domain U ⊂K^r, (t1, . . . ,tr)∈ U, polynomialsgi,α(t, z)∈Q(t)[z], for everyi and α, andz(t)∈QA(U) such thatg_i,α(t, z(t)) =g_i,α for everyiandα.

Since the al and the ej are rational functions with rational coefficients in the gi,α, the system of equations (5.3) is equivalent to a system of polynomial equations with rational coefficients

(5.4) f(g_i,α) = 0.

By Theorem 3 the functionsgi,α(t, z(t)) are solutions of this system of equations.

Because ej(gi,α) = ej(gi,α(t, z(t))) 6= 0 for every j, we have that none of the ej(gi,α(t, z(t))) vanishes in a small open ballV ⊂ U centered att.

In particular we have that

(5.5) e_j(g_i,α(t, z(t)))6= 0 ∀j, ∀t∈ V.

Fort∈ V andr= 1, . . . , s, we define g⁰_r(t, x) := X

α∈Nⁿ

gi,α(t, z(t))x^α,

Fn(t, x) =

s

Y

r=1

Gr(t, x), where G_r(t, x) = X

α∈Nⁿ

A_n,r,α(g_i,α(t, z(t)))x^α, r= 1, . . . , s, and forj=k0, . . . , n−1

Fj(t, x) = X

α∈Nⁿ

Aj,α(gi,α(t, z(t)))x^α. In particular we have thatGr(t, x) =ϕµ(gr(x)) for everyr.

Let us denote byOVthe ring ofK-analytic functions onV. In particular we have thatF_j(t, x) is a polynomial ofOV[xⁱ], of degreed_j inx_j, such that the coefficient of x^d_j^j is ej(gi,α(t, z(t))) and whose discriminant (seen as a polynomial in xj) is equal toej−1(gi,α(t, z(t)))Fj−1(t, x)∈ OV[x^j−1] for everyj by (5.4).

Thus the family (F_j)_j satisfies the hypothesis of Theorem 5 by (5.5). Hence the algebraic hypersurfacesX₀:={Fn(0, x) = 0}andX₁:={Fn(1, x)}are homeomor- phic. Moreover the homeomorphism between them maps every component of X₀ defined byG_r(0, x) = 0 onto the component of X₁ defined byG_r(1, x) = 0. This proves that the algebraic variety defined by{ϕ_µ(g₁) =· · ·=ϕ_µ(g_s) = 0} is home- omorphic to the algebraic variety{G1(0, x) =· · ·=Gs(0, x) = 0} which is defined by polynomial equations overQ. ThusV is homeomorphic toW :={g₁⁰(x) =· · ·= g_s⁰(x) = 0}.

Moreover since Q∩K is dense in K, we can choose q ∈ Q as close as we want to t. In particular by choosing q close enough to t we may assume that (iv) in Theorem 10 is satisfied, sinceγ,t7→z(t) and theg_i,αare continuous functions.

Finally we have that z(q) is algebraic of degree ≤ d over Q by Lemma 4 (see Remark 11).

Thus Theorem 10 is proven in the affine case. The projective is proven is the same

way by Remark 6.

We can remark that Theorem 10 (v) implies that several algebraic invariants are preserved by the homeomorphismh. For instance we have the following corollary:

Corollary 13. For ε > 0 small enough the Hilbert-Samuel function of W is the same as that ofV.

Proof of Corollary 13. Let h₁, . . . , h_m be a Gr¨obner basis of the ideal I of K[x]

generated by theg_iwith respect to a given monomial order. Let us recall that for f =P

α∈Nⁿf_αx^α∈K[x],f 6= 0, we denote the leading term off by LT(f) =f_α0x^α0 where α₀ is the largest nonzero exponent of the support of f with respect to the monomial order:

Supp(f) ={α∈Nⁿ|fα6= 0}.

A Gr¨obner basis ofIis computed by considering S-polynomials and divisions (see [CLO07] for more details):

- the S-polynomial of two nonzero polynomials f and g is defined as follows: set LT(f) = ax^α and LT(g) = bx^β with a, b ∈ K^∗, and let x^δ be the least common multiple ofx^α andx^β. Then the S-polynomial off andg is

S(f, g) := x^δ−α

a f−x^δ−β b g.

- the division of a polynomialf by polynomialsf1, . . . ,flis defined inductively as follows: firstly after some renumbering one assume that LT(f1)LT(f2) · · · LT(fl). Then we consider the smallest integer i such that LT(f) is divisible by LT(f_i). If such aiexists one setsq_i⁽¹⁾= LT(f)

LT(fi) andq⁽¹⁾_j = 0 forj6=iandr⁽¹⁾ = 0.

Otherwise one sets q_j⁽¹⁾ = 0 for every j and r⁽¹⁾ = LT(f). We repeat this process by replacingf byf −Pl

j=1q_j⁽¹⁾fj−r⁽¹⁾. After a finite number of steps we obtain a decomposition

f =

l

X

j=1

qjfj+r

where none of the elements of Supp(r) is divisible by any LT(fi). The polynomial ris called the remainder of the division off by thefi.

Thus we can make the following remark: every remainder of the division of some S-polynomialS(gi, gj)byg1, . . . ,gs is a polynomial whose coefficients are rational functions on the coefficients of the gk.

The Buchberger’s Algorithm is as follows: we begin with g1,. . . , gs the generators ofIand we compute all the S-polynomials of every pair of polynomials among the gi. Then we consider the remainders of the divisions of these S-polynomials by the gi. If some remainders are nonzero we add them to the family {g1, . . . , gs}. Then we repeat the same process that stops after a finite number of setps.

By the previous remark ifhdenotes the remainder of the division of a S-polynomial S(g_i, g_j) by theg_k then we have a relation of the form

(5.6) S(gi, gj) =

s

X

l=1

blgl+h

and the bl belong to the field extension of Q generated by the coefficients of the gl. Lethβ denote the coefficient of x^β in h. Then for those hβ that are nonzero, Equation (5.6) provides an expression of them as rational functions in thegi,α, let us say

(5.7) ∀β∈Supp(h) h_β=H_β(g_i,α).

For those hβ equal to zero Equation (5.6) provides a polynomial relation with rational coefficients between thegi,α:

(5.8) ∀β /∈Supp(h) Hβ(gi,α) = 0.

And Equation (5.6) is equivalent to the systems of equations (5.7) and (5.8). This remains true if we replace the gi by polynomials whose coefficients are rational functions in thegi,α with rational coefficients. Thus the fact that h1,. . . , hm is a Gr¨obner basis obtained from theg_i by Buchberger’s Algorithm is equivalent to a system of equations

(5.9) Qk(gi,α) = 0 for k∈E

whereEis a (not necessarily finite) set. Thus by Theorem 10 (v) we see that these equations are satisfied by the coefficients of theg⁰_i, hence the ideal definingW has a Gr¨obner basis h⁰₁, . . . , h⁰_m obtained from theg_i⁰ by doing exactly the same steps in Buchberger’s Algorithm, and Supp(h⁰_i)⊂Supp(h_i) for everyi.

Moreover the initial terms of theh_i are rational functions in theg_i,αwith rational coefficients, let us sayHi(gi,α) for some rational functionsHi. By choosingεsmall enough we insure that

H_i(g_i,α⁰ )6= 0 ∀i.

Thus the leading exponents of theh⁰_i are equal to those of theh_i. In particular the Hilbert-Samuel function ofW is the same as that ofV, and (iv) in Theorem 10 is

proven.

Remark 14. In fact we have proven that the ideal of leading terms ofIis the same as the ideal of leading terms of the ideal definingW. So we could have concluded by [GP08, Theorem 5.2.6] for instance.

6. Complete algorithm 6.1. Settings.

Input:

(1) polynomials g1, . . . , gs ∈ K[x] whose coefficients gi,α belong to a finitely generated field extensionkoverQ.

(2) a presentation

k=Q(t1, . . . ,tr,z)

where the ti are algebraically independent over Q and z is finite over Q(t1, . . . ,tr).

(3) the minimal polynomialP(z) ofzoverQ(t1, . . . ,tr).

(4) the gi,αare given as rational functions in theti andz, and the coefficients ofP(z) as rational functions in theti.

(5) a positive real numberε.

Moreover we assume that the t_i and z are computable numbers [Tu36]. Let us recall that a real computable number is a numberx∈Rfor which there is a Turing machine that computes a sequence (q_n) of rational numbers such that|x−q_n|< _n¹ for everyn≥1. A computable number is a complex number whose real and imag- inary parts are real computable numbers. The set of computable numbers is an algebraically closed field [Ri54]. More precisely ift1,. . . , tr are computable num- bers such that for everyi∈ {1, . . . , r} (qi,n)n is a sequence ofQ+iQcomputed by a Turing machine with|ti−q_i,n|<_n¹ for everyn, and ifz∈CsatisfiesP(t,z) = 0

for some reduced polynomialP(t, z)∈Q[t, z], then one can effectively find a Turing machine that computes a sequence (qn)nofQ+iQsuch that|z−qn|< _n¹ for every integern.

Output: polynomialsg₁⁰,. . . ,g_s⁰ ∈(Q∩K)[x] with the properties:

(1) the pairs (V(g_i),Kⁿ) and (V(g_i⁰),Kⁿ) are homeomorphic, and the homeo- morphism is subanalytic and arc-analytic.

(2) the coefficientsg_i,α⁰ ofg⁰_i satisfy the properties:

g_i,α⁰ 6= 0⇐⇒gi,α6= 0

|g_i,α⁰ −gi,α|< εfor everyi andα

and every polynomial relation with coefficients inQsatisfied by thegi,αis also satisfied by theg_i,α⁰ .

6.2. Algorithm: We present here the successive steps of the algorithm.

(1) We make a linear change of coordinates with coefficients in Q, denoted by ϕµ withµ∈Qⁿ⁻¹, such that each of thegi is a monic polynomial inxn of degree deg(gi).

We denote byf_n the product of theg_i after this change of coordinates, and bya_n the vector of the coefficients of the g_i after this change of coordinates (seen as a polynomial inx_n).

(2) For everyj from n to 1 we do the following: let us assume thatf_j is a poly- nomial in x1, . . . , xj having a nonzero monomialejx^deg(f_j ^j), ej ∈ k^∗. We denote byaj the vector of the coefficients of fj seen as a polynomial in xj. We consider the generalized discriminants ∆j,l offj with respect toxj, and we denote bylj the smallest integer such that

∆j,l_j(aj)6= 0.

These polynomials ∆j,l_j(aj) can be effectively computed (see 6.3).

We perform a linear change of coordinates (with coefficients inQ) in x1, . . . ,x_i−1 such that ∆j,l_j(aj) becomes a unit times a monic polynomial of degree deg(∆j,l_j(aj)) inx_j−1, and we denote byf_j−1 this new monic polynomial.

(3)We stop the process once we have thatfj−1 is a nonzero constant.

(4) We consider (5.2) as a system of polynomial equations that will allow us to compute the expression of thee_j as rational functions in theg_i,α.

(5)We denote byP(t, z) the monic polynomial ofQ(t)[z] such thatP(t, z) =P(z) is the minimal polynomial ofzoverQ(t).

By replacingεby a smaller positive number we may assume that d(t,∆_P)>2ε

where ∆P denotes the discriminant locus ofP(t, z) seen as a polynomial inz. This discriminant is computed as Resz(P, ∂P/∂z). See [DGY96] or [BM95, Theorem C]

for a practical way of choosing such aε.

(6)Letz₁:=z, z₂,. . . ,z_d be the distincts roots of P(t, z). These are computable

numbers and so we can computeη ∈Q>0such that all the differences between two zi are strictly greater thanη.

Letz(t) be the root ofP(t, z) such thatz(t) =z. We can write w(t) :=z(t+t) = X

α∈N^r

zαt^α, z0=z.

We write

P(t, z) =p0(t) +p1(t)z+· · ·+pd−1(t)z^d−1+z^d where thepi(t)∈K(t). Set

M := 1 + max

0≤i≤d−1 max

t⁰∈B(0,2ε)|pi(t⁰)|.

Then |zα| ≤ M

(2ε)^|α| for every α (by the Cauchy bounds for the roots of a monic polynomial since w(t) is convergent on B(0,2ε) by (5)). Let W_k(t) be the homo- geneous term of degree k in the Taylor expansion of w(t): W_k(t) =P

|α|=kz_αt^α. Then

∀t⁰∈B(0, ε) |Wk(t⁰)| ≤ M 2^k

k+r−1 r−1

. We have that

∀k≥r M 2^k

k+r−1 r−1

≤ M2^r (r−1)!

k^r 2^k. Then choosek₀≥rsuch that k^r

√ 2^k

≤1 forallk≥k₀. Therefore

∀t⁰∈B(0, ε),∀k≥k0 |Wk(t⁰)| ≤ M2^r (r−1)!

√1 2^k and

∀t⁰ ∈B(0, ε),∀k≥k0

X

l≥k

Wl(t⁰)

≤ M2^r (r−1)!

√

√ 2 2−1

√1 2^k

=C 1

√2^k .

(7) Now we want to determine the computable number z(q) = w(q−t) for a given choice ofq∈B(t, ε)∩(Q+iQ)^r. This number is one of the roots ofP(q, z).

These roots are computable numbers and so we can bound from below all the dif- ferences between each two of them: letδ∈Q>0be such a bound.

By the previous step one can compute an integerksuch that

∀t⁰∈B(0, ε),

X

l≥k

Wl(t⁰)

≤C 1

√ 2^k

≤ δ 2. So we can distinguishz(q) from the other roots ofP(q, z).

(8)Chooseq∈(Q+iQ)^rsuch that

(i) qis not in the discriminant locus ofP(t, z) seen as a polynomial inz, (ii) kq−tk< ε,

(iii) e_j(g_i,α(q, z(q)))6= 0.

The first condition is insured by the choice ofεin (5).

In order to check that ej(gi,α(q, z(q))) 6= 0, one only has to choose q such that kt−qk is small enough and this can be done effectively. Indeed theej(gi,,α) are rational functions in the t_i and z, thus we can effectively bound the variations of e_j locally aroundt.

(9) Then we evaluate the g_i,a(t, z(t)) at (q, z(q)). We denote these values by g_i,α⁰ and we define the polynomials

g_i⁰:= X

α∈Nⁿ

g⁰_i,αx^α.

6.3. Generalized discriminants. We follow Appendix IV of [Wh72] or [Roy06].

The generalized discriminant ∆l of a polynomial f =x^d+

d

X

j=1

bjx^d−j

can be defined as follows:

Letξ1,. . . ,ξdbe the roots off (with multiplicities), and setsi =Pd

k=1ξⁱ_kfor every i∈N. Then

∆_d+1−l=

s0 s1 · · · sl−1

s1 s2 · · · sl

· · · · sl−1 sl · · · s2l−2

.

Thus ∆₁ is the classical discriminant off. The polynomials ∆_lmay be effectively computed in term of the s_i, and those can be effectively computed in terms of the b_i. The polynomial f admits exactly k distinct complex roots if and only if

∆1=· · ·= ∆_d−k = 0 and ∆_d−k+16= 0.

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E-mail address:adam.parusinski@unice.fr

Universit´e Nice Sophia Antipolis, CNRS, LJAD, UMR 7351, 06108 Nice, France E-mail address:guillaume.rond@univ-amu.fr

Aix Marseille Univ, CNRS, Centrale Marseille, I2M, Marseille, France