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 ⊂K^{n} (resp. V ⊂P^{n}_{K}) be an affine (resp. projective) algebraic
set, where K = R or C. Then there exist an affine (resp. projective) algebraic
set W ⊂ K^{n} (resp. W ⊂ P^{n}_{K}) and a homeomorphism h : K^{n} −→ K^{n} (resp.

h:P^{n}_{K}−→P^{n}_{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,α}is 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), x^{i} 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^{r}be 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(t_{1}, . . . ,t_{r}).

Then there exist new variables t= (t_{1}, . . . , 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= (y_{1}, . . . , 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 t_{1}, . . . , 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}(t_{1}, . . . ,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(t_{1}, . . . ,t_{r},z) = 0.

Sett:= (t1, . . . ,tr)∈K^{r}and 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. Thenw_{1}∈QA(U) and the Taylor series of
w_{1}at 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:= (b_{1}, . . . , b_{r})∈Q^{r}_{K}∩ U \{q= 0}. Let us denote byϕ_{1}(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) := (y_{1}(t, z, v), . . . , y_{m}(t, z, v)).
Letf(y)∈Q[y]^{p} such thatf(y) = 0. We have that

f(y(t_{1}, . . . ,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 w_{1}. Let P_{1} ∈Q[t_{1}, . . . , t_{r}, z] be a polynomial
of minimal degree such thatP_{1}(t, ϕ(t)) = 0. Note thatP_{1}is irreducible. Write

P_{1}(t, z) =

degP_{1}

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, ϕ_{2}(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^{r}be 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 inK^{n}. Suppose that
fori=k_{0}, . . . , n, there are given

Fi(t, x^{i}) =x^{d}_{i}^{i}+

d_{i}

X

j=1

a_{i−1,j}(t, x^{i−1})x^{d}_{i}^{i}^{−j}∈ O_{V}[x^{i}],

with di>0, such that

(i) for every i > k_{0}, the first non identically equal to zero generalized dis-
criminant ofF_{i}(t, x^{i−1}, x_{i})equalsF_{i−1}(t, x^{i−1})up to a multiplication by a
nowhere vanishing function ofO_{V}.

(ii) the first non identically zero generalized discriminant ofF_{k}_{0} is independent
of xand does not vanish at t.

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

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

Moreover ifFn=G1· · ·Gs then for every j= 1, . . . , s
h_{q} G^{−1}_{j} (0)∩({t} ×K^{n})

=G^{−1}_{j} (q)∩({0} ×K^{n}).

Remark 6. The homeomorphismh_{q}of Theorem 5 is written ash_{q}(t, x) = (q,Ψ_{q}(x))
for everyx. In the case where theF_{i}are homogeneous polynomials with respect to
the indeterminatesx, the homeomorphisms Ψ_{q} satisfy

∀λ∈K^{∗},∀x∈K^{n} Ψq(λx) =λΨq(x)

Hence if we defineP(V_{q}) ={(q,x)∈ V ×P^{n}_{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^{r}and a homeomorphism

Φ :W ×K^{n}→ W ×K^{n}

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)→K^{n},
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})∈Q^{n−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
byx_{1}, . . . , x_{n−1}. So we have that

ϕµ(f) =cx^{d}_{n}+

d

X

j=1

aj(x^{n−1})x^{d−j}_{n}
for some polynomialsa_{j}(x^{n−1})∈k[x^{n−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}_{n}^{i}+

d

X

j=1

ai,j(x^{n−1})x^{d−j}_{n}
for some nonzero constantsci∈k.

Remark 9. For everyf ∈k[x] of degree dwhere k⊂K andµ∈Q^{n−1}, we have
that

ϕ−µ◦ϕµ(f) =f.

Let us define the support off =P

α∈N^{n}f_{α}x^{α} as

Supp(f) :={α∈N^{n} |f_{α}6= 0}.

Let us assume that

ϕµ(f) =X
f_{α}^{0}x^{α}

for some polynomialsf_{α}^{0} ∈k. Then the coefficientfα has the form
fα=Pα(µ1, . . . , µn−1, f_{β}^{0})

for some polynomialPα∈Z[µ, f_{β}^{0}] depending only on thef_{β}^{0} with|β|=|α|.

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

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

h:P^{n}_{K}−→P^{n}_{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
polynomialsg_{1}^{0},. . . ,g_{s}^{0} ∈Q[x] such that if we write

g_{i}= X

α∈N^{n}

g_{i,α}x^{α} andg_{i}^{0}= X

α∈N^{n}

g^{0}_{i,α}x^{α}

then for every iandαwe have that:

|gi,α−g^{0}_{i,α}|< ε.

(v) Every polynomial relationship with rational coefficients between thegi,αwill
also be satisfied by theg_{i,α}^{0} .

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_{i}^{0} 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,α}^{0} = 0.

Ifεis chosen small enough we may even assume that

∀i, α gi,α= 0⇐⇒g^{0}_{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∈Q^{n−1} as
in Section 4 we can assume that

ϕ_{µ}(g_{r}) =c_{r}x^{p}_{n}^{r}+

p_{r}

X

j=1

b_{n−1,r,j}(x^{n−1})x^{p}_{n}^{r}^{−j} = X

β∈N^{n}

a_{n,r,β}x^{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)^{N}^{n} 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 x^{n−1} with coefficients in Q we can write

∆n,l_{n}(an) =en−1fn−1 with
fn−1= X

β∈N^{n}

an−1,βx^{β}=x^{d}_{n−1}^{n−1}+

dn−1

X

j=1

bn−2,j(x^{n−2})x^{d}_{n−1}^{n−1}^{−j}

for some constantse_{n−1}, a_{n−1,β}∈k, withe_{n−1}6= 0, and some polynomialsb_{n−2,j} ∈
k[x^{n−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,l}_{n−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,l}_{j+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∈N^{n}

a_{j,β}x^{β}

=e_{j}f_{j}
is the first non identically zero generalized discriminant of fj+1, whereaj denotes
the vectors of coordinatesaj,β in k^{N}^{j}, i.e. we have that :

(5.2) ∆j+1,l_{j+1}(aj+1) =ej

X

β∈N^{n}

aj,βx^{β}

=ejfj

and

(5.3) ∆j+1,l(aj+1)≡0 ∀l < lj+1,
until we getf_{k}_{0} = 1 for somek_{0}≥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)^{N}^{k} 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
g^{0}_{r}(t, x) := X

α∈N^{n}

gi,α(t, z(t))x^{α},

Fn(t, x) =

s

Y

r=1

Gr(t, x), where
G_{r}(t, x) = X

α∈N^{n}

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∈N^{n}

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[x^{i}], 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 hypersurfacesX_{0}:={Fn(0, x) = 0}andX_{1}:={Fn(1, x)}are homeomor-
phic. Moreover the homeomorphism between them maps every component of X_{0}
defined byG_{r}(0, x) = 0 onto the component of X_{1} defined byG_{r}(1, x) = 0. This
proves that the algebraic variety defined by{ϕ_{µ}(g_{1}) =· · ·=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 :={g_{1}^{0}(x) =· · ·=
g_{s}^{0}(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,α}are 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 h_{1}, . . . , h_{m} be a Gr¨obner basis of the ideal I of K[x]

generated by theg_{i}with respect to a given monomial order. Let us recall that for
f =P

α∈N^{n}f_{α}x^{α}∈K[x],f 6= 0, we denote the leading term off by LT(f) =f_{α}_{0}x^{α}^{0}
where α_{0} is the largest nonzero exponent of the support of f with respect to the
monomial order:

Supp(f) ={α∈N^{n}|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 theg^{0}_{i}, hence the ideal definingW has
a Gr¨obner basis h^{0}_{1}, . . . , h^{0}_{m} obtained from theg_{i}^{0} by doing exactly the same steps
in Buchberger’s Algorithm, and Supp(h^{0}_{i})⊂Supp(h_{i}) for everyi.

Moreover the initial terms of theh_{i} are rational functions in theg_{i,α}with rational
coefficients, let us sayHi(gi,α) for some rational functionsHi. By choosingεsmall
enough we insure that

H_{i}(g_{i,α}^{0} )6= 0 ∀i.

Thus the leading exponents of theh^{0}_{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}|< _{n}^{1}
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}|<_{n}^{1} 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|< _{n}^{1} for every
integern.

Output: polynomialsg_{1}^{0},. . . ,g_{s}^{0} ∈(Q∩K)[x] with the properties:

(1) the pairs (V(g_{i}),K^{n}) and (V(g_{i}^{0}),K^{n}) are homeomorphic, and the homeo-
morphism is subanalytic and arc-analytic.

(2) the coefficientsg_{i,α}^{0} ofg^{0}_{i} satisfy the properties:

g_{i,α}^{0} 6= 0⇐⇒gi,α6= 0

|g_{i,α}^{0} −gi,α|< εfor everyi andα

and every polynomial relation with coefficients inQsatisfied by thegi,αis
also satisfied by theg_{i,α}^{0} .

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∈Q^{n−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)Letz_{1}:=z, z_{2},. . . ,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

t^{0}∈B(0,2ε)|pi(t^{0})|.

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_{α}t^{α}.
Then

∀t^{0}∈B(0, ε) |Wk(t^{0})| ≤ 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 choosek_{0}≥rsuch that k^{r}

√
2^{k}

≤1 forallk≥k_{0}. Therefore

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

√1
2^{k}
and

∀t^{0} ∈B(0, ε),∀k≥k0

X

l≥k

Wl(t^{0})

≤ 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

∀t^{0}∈B(0, ε),

X

l≥k

Wl(t^{0})

≤C 1

√
2^{k}

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

(8)Chooseq∈(Q+iQ)^{r}such 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,α}^{0} and we define the polynomials

g_{i}^{0}:= X

α∈N^{n}

g^{0}_{i,α}x^{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=1ξ^{i}_{k}for 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 ∆_{l}may 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+1}6= 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