Cell decomposition for two dimensional local fields

Cell Decomposition for Two Dimensional Local Fields.

ALIBLEYBEL(*)

ABSTRACT- We prove a cell decomposition theorem for the two-dimensional local fieldQ_p((t)).

Introduction.

The purpose of this paper is to give a cell decomposition for the field of Laurent series on p-adic fields. Originally, cell decomposition theorems were used to prove rationality of Igusa Zeta function, and of Poincare series, avoiding resolution of singularities. Also, using cell decomposition for a p-adic field (forp a fixed prime), Denef (1986) gave a new proof of elimination of quantifiers inQ_p. Later, Pas (1989) proved a uniform (inp) cell decomposition for p-adic fields, thus obtaining auniform quantifier elimination.

In this paper we prove a cell decomposition for Q_p((t)) (or for K((t)) whereKis a finite algebraic extension ofQ_p), using both thet-adic andp- adic valuations. Our proof uses a mixture of Denef and Pas results, and may be useful for the development of motivic integration on the two-di- mensional local fieldQ_p((t)), in a similar way to the work of Cluckers &

Loeser (2004) (hereafter CL 2004). Note that integration over higher di- mensional local fields was also addressed by Hrushovski & Kazhdan (2005).

In section 1 we recall the language of 2-valued fields used in the paper, as well of the definitions for definable sets and cells. In section 2 we state Cell Decomposition I together with its proof. This section also contains a lemma needed in the proofs of theorems I and II. Theorem II (Cell de- composition II) is stated and proved in section 3. Finally a generalized cell decomposition theorem is stated and proved in section 4.

(*) Indirizzo dell'A.: Lebanese University, Faculty of Sciences, Beirut, Lebanon E-mail: [email protected]

Acknowledgments. I gratefully thank Francois Loeser who suggested the problem and pointed out to me the basic idea for the solution. I also thank Raf Cluckers for fruitful discussions, and an anonymous referee for constructive comments.

1. Language of 2-valued fields.

An n-dimensional local field is a complete discrete valuation field F whose residue field isn 1-dimensional local field (Fesenko 2003). LetK be a 2-dimensional local field, also named 2-valued field.K is equipped with a valuation ord1: K!G for some ordered abelian group G, R its va- luation ring with residue field K.K is also a valued field, with valuation ord₂:K !SwithSan ordered abelian group,Rits valuation ring andk the residue field. There is a projection res: R!KˆR= Pof the valuation ring onto the residue field, where P is the maximal ideal ofR. We also define an angular component map ac: K!K. ac is a multiplicative map which can be extended by putting ac(0)ˆ0. The language adopted is a multi-sorted language

L ˆ(L_Val;L_RV;L_Ord1;L_Ord2;ord₁;ord₂;ac) with 4-sorts

(i) a Val-sort for the 2-valued field sort, (ii) an RV-sort for the 1-valued field sort,

(iii) an Ord₁-sort for the value group(with respect to the first va- luation map) sort, and

(iv) an Ord₂-sort for the value group(with respect to the second valuation map) sort,

where LVal is the language of rings LRingsˆ(‡; ; :;0;1) and LRV is an expansion of Macintyre's language LMacLRings[ fPnjn2N;n>1g wherePnare predicates whose interpretations are the set of nonzeron^th- power) andL_Ord1 (andL_Ord2) is an expansion of the language of ordered groups, for instance L_PR1, a variant of the Presburger language LPR f‡;0;1;g [ fnjn2N;n>1gdefined byLPR1 LPR[ f1g. A structure for this language consists of a tuple (K; K;G;S) whereK is a 2- valued field with residue fieldK, value groupGvaluation mapord1and an angular component map ac, andKis a valued field with a value groupSand a valuation mapord₂, together with an interpretation ofL_RVinK, and an interpretation ofL_Ord1andL_Ord2inGandSrespectively.

A formula in this language is built upfrom symbols ofLtogether with variables, the logical connectives^(and),_(or),:(not), the quantifiers9, 8, the equality symbolˆand parameters.

When K ˆK((t)) for some valued field K, there exists a natural va- luation mapord₁:ˆord_t:K((t))!Z(extended by putting ord_t(0)ˆ 1) and a natural angular component map ac:P

ilaitⁱ7!al 6ˆ0. The valued field K is assumed Henselian of zero characteristic.

If we interpret_ninL_Ord1andL_Ord2as ``congruent modulon'' inZthen (Q_p((t));Q_p;Z;Z) is a structure for the languageL, with the naturalp- adic valuation ordpas an interpretation for ord2(where we also extend ordp

by ordp(0)ˆ 1).

Also, note that the mapres is definable in this language (in a non-ca- nonical way) (see the remark under definition 2.2 and the lemma 3.4 in (Pas 1989)).

Notice the analogy of the languageLwith the three-sorted Denef-Pas languageLDP(Pas 1989).

Finally, we will use the fact that if (K; K;G;S) is a structure forLthen (K;S) is a structure for the 2-sorted languageL_Macˆ(L_RV;L_Ord2).

Consider now the L-theory T₂ of 2-valued Henselian fields of zero characteristic and having surjective valuation maps, surjective angular component map, and 1-valued Henselian field of characteristic zero and bounded ramification (ord₂(p)ˆ1 for some primep).

In this context we have the following variant of Denef-Pas Theorem on elimination of 2-valued field quantifiers

THEOREM1.1. The theoryT2admits elimination of quantifiers in the 2-valued field sort. More precisely,every L formula f(x;j;a;b),with x variables in theVal-sort,jvariables in theRV-sort,avariables in theOrd₁- sort andbvariables in theOrd₂-sort,isT₂equivalent to a finite disjunction of formulas of the form

c(acf1(x);. . .;acf_k(x);j;b)^u(ord1f1(x);. . .;ord1f_k(x);a;b);

withc a(L_RV[L_Ord2)-formula,u a(L_Ord1[L_Ord2)-formula and f₁;. . .;f_k polynomials inZ[X].

Note the analogy of our statement with theorem 2.1.1 of (CL 2004).

PROOF. Direct application of theorem 4.1 and lemma 5.3 of Pas (1989).

p

A subsetCofK^mKⁿG^r1S^r2(wherem;n;r1 andr2are positive integers) is called definable if it is definable by anL-formula.

A functionf is definable if its graph is a definable subset.

A subsetDofKⁿS^r isL_Mac-definable if it is definable by an L_Mac- formula.

AnLMac-definable subset ofKⁿ(whenKis ap-adic field, that is a finite algebraic extension of Q_p) is also a semi-algebraic set in the standard terminology, e.g. Denef (1984).

REMARK1.2. It should be noted that even ifGˆSˆZ, it is not al- lowed to mix variables ofG-sort with variables ofS-sort.

DEFINITION1.3. Letxˆ(x1;. . .;xm) be Val-variables,jˆ(j₁;. . .;j_n) RV-variables. LetCbe a definable subset ofK^mKⁿ. Letb₁,b₂,cbe de- finable functions fromC toK, l₁,l₂ positive integers,d₁,d₂,edefinable functions from Proj_KnC(the image ofCby the projection ofK^mKⁿonto Kⁿ) toK, and let}1,}2,p1andp2be<,, or no condition.

For eachj2Kⁿ, letA(j) be the set of (x;T)2K^mK subject to the definable conditions

n(x;T)2K^mKj(x;j)2C;ord₁b₁(x;j)}₁l₁ord₁(T c(x;j))}₂ord₁b₂(x;j);

ord₂d₁(j)p₁l₂ord₂(ac(T c(x;j)) e(j))p₂ord₂d₂(j)o and suppose that for allj;j⁰2Kⁿwithj6ˆj⁰A(j)\A(j⁰)ˆ , then

Aˆ[

j

A(j)

is acellinK^mKⁿwith parameters (j₁;. . .;j_n), primary centerc(x;j) and secondary centere(j);A(j) is afiberof the cellA.

2. Cell decomposition I.

Now consider a model for the theory T₂ such that K ˆK((t)) and GˆSˆZ.

THEOREM 2.1. Let f(x;T) be a polynomial in T of degree d whose coefficients are definable functions in x2K((t))^m;j2Kⁿ.

Then there exists a partition of K((t))^mK((t))in a finite number of cells A with parameters(j₁;. . .;j_n;j⁰₁;. . .;j⁰_r)2K^n‡r,each cell has primary and

secondary centers c(x;j)and e(j;j⁰)respectively such that if we write f(x;T)ˆX^d

iˆ0

a_i(x;j)(T c(x;j))ⁱ (2:1:1)

then for all(j;j⁰)2K^n‡rand for all(x;T)2A(j;j⁰)we have (2:1:2) ord_tf(x;T)ˆord_ta_i0(x;j)(T c(x;j))ⁱ⁰ˆmin

0idord_ta_i(x;j)(T c(x;j))ⁱ where i₀does not depend on(x;j;j⁰;T),and

(2:1:3) ord_p(acf(x;T)) min

0idord_p

b_i(j;j⁰) ac(T c(x;j)) e(j;j⁰)_i

‡l for some l2N; b_i(j;j⁰)are (partial)LMac-definable functions K^n‡r!K (to be defined below) and j⁰₁;. . .;j⁰_r2K such that j⁰₁ˆacf₁(x;j);. . .;j⁰_rˆ

ˆacf_r(x;j) where f₁(x;j);. . .;f_r(x;j) are polynomials in x with integer coefficients.

PROOF. We assume the result for all polynomials of degree<d. The theorem then holds forf⁰(x;T) (derivative off(x;T) with respect toT), and so there exists a partition ofK((t))^mK((t)) in cellsAˆS

j;j⁰

A(j) (of centers c(x;j) ande(j;j⁰)) as above such that

ord_t f⁰(x;T)ˆ min

1idord_tia_i(x;j)(T c(x;j))^{i 1} (2:1:4)

and

(2:1:5) ordp(acf⁰(x;T)) min

0id 1ord_p

b⁰_i(j;j⁰) ac(T c(x;j)) e(j;j⁰)_i

‡l⁰ whereb⁰_i(j;j⁰) are partial definable functionsK^n‡r!K,l⁰2N,j⁰_jˆf_j⁰(x;j) andf_j⁰(x;j) are polynomials inxwith integer coefficients.

For eachj2Kⁿ, letA(j) be defined by

n(x;T)2K((t))^mKj(x;j)2C;ordtb1(x;j)}1l1ordt(T c(x;j))}2ordtb2(x;j);

ord_pd₁(j)p₁l₂ord_p(ac(T c(x;j)) e(j))p₂ord_pd₂(j)o wherel1;l2;C;b1;b2;d1;d2;candeare as in (1.3) and where the RV-vari- ablesj⁰were added to the parametersj(by defining e.g.c⁰(x;j;j⁰)c(x;j) for all (x;j;j⁰)2K((t))^mKⁿK^r and replacing n‡r by n⁰, and then renamingc⁰ascandn⁰asn).

We will further partitionAinto subcells on which the theorem holds for f(x;T).

Consider only the nontrivial case, where the set~Iofjsuch that ordtaj(x;T)(T c(x;j))^jˆ min

0idordtai(x;j)(T c(x;j))ⁱ (2:1:6)

has a cardinality>1.

Leti₀2~I; then each fiberA(j) is a disjoint union of two sets (2:1:7) A1(j)ˆ

(x;T)2A(j)jordt f(x;T)ˆordtai0(x;j)(T c(x;j))ⁱ⁰ and

(2:1:8) A2(j)ˆ

(x;T)2A(j)jordt f(x;T)>ordta_i0(x;j)(T c(x;j))ⁱ⁰ : Consider the polynomialsFj(z) given by

F_j(z)ˆX

i2~I

g_ij(j;j⁰)zⁱ (2:1:9)

wheregijareLMac-definable functionsKⁿK^r!K, such that ac(a_i(x;j))ˆg_ij(j;acf1(x;j);. . .;acfr(x;j))

for (x;j)2X_j, where (X_j)_jform a partition ofK((t))^mKⁿ, andf₁;. . .;f_rare polynomials inxwith integer coefficients, as in lemma 2.2 below.

As the functions gij are definable in LMac, we can apply the cell de- composition theorem I of Denef (1986) toFj(z); it follows that there exists a finite partition of K^n‡r‡1 in Denef-type cells B_j (of center e_j(j;j⁰), (j;j⁰)2K^n‡r) defined by

B_jˆ

(j;j⁰;z)2K^n‡rKj(j;j⁰)2D_j;ordpc_1j(j;j⁰)p_1j

ord_p(z e_j(j;j⁰))p_2jord_pc_2j(j;j⁰) ; whereD_jK^n‡ris aLMac-definable set, andc_1j(j;j⁰),c_2j(j;j⁰) ande_j(j;j⁰) are

L_Mac-definable functionsK^n‡r!K, such that ordpFj(z) min

0idordp(bij(j;j⁰)(z ej(j;j⁰))ⁱ)‡lj

(2:1:10)

withl_j 2N;b_ij(j;j⁰) are the coefficients ofF_j(z) written in the form F_j(z)ˆX

i

b_ij(j;j⁰)(z e_j(j;j⁰))ⁱ; (2:1:11)

andp_1j;p_2j(for eachj) is<;or no condition.

Aszˆej(j;j⁰) if and only if ordp(z ej(j;j⁰))ˆ 1, it follows that by the above observation, (x;T)2A2if and only if

F_j(ac(T c(x;j))ˆ0 for somej, if and only if

ac(T c(x;j))ˆe_j(j;acf₁(x;j);. . .;acf_r(x;j)) andb0j(j;acf1(x;j);. . .;acfr(x;j))ˆ0.

It follows thatA1(j) can be written as a finite union of sets of the fol- lowing form

[

j⁰

n(x;T)2Aj(x;j;j⁰)2Y_j;ac(T c(x;j))6ˆe_j(j;j⁰);

ordpc1j(j;j⁰)p1jordp(ac(T c(x;j)) ej(j;j⁰))p2jordpc2j(j;j⁰)o or

[

j⁰

n(x;T)2Aj(x;j;j⁰)2Z_j;b_0j(j;j⁰)6ˆ0;ac(T c(x;j))ˆe_j(j;j⁰)o : Also it follows thatA2(j) is a finite union of sets of the form

[

j⁰

n(x;T)2Aj(x;j;j⁰)2Z_j;b_0j(j;j⁰)ˆ0;ac(T c(x;j))ˆe_j(j;j⁰)o

;

where

Y_jˆ f(x;j;j⁰)2K((t))^mKⁿK^rj(x;j)2X_j;(j;j⁰)2D_j; acf1(x;j)ˆj⁰₁;. . .;acfr(x;j)ˆj⁰_rg;

and

Z_jˆ f(x;j;j⁰)2K((t))^mKⁿK^rj(x;j)2X_j;(j;j⁰)2D_j; acf1(x;j)ˆj⁰₁;. . .;acfr(x;j)ˆj⁰_r;c_1j(j;j⁰)_1j0;c_2j(j;j⁰)_2j0g with1jis6ˆor no condition and2j isˆor no condition.

Note that one of the centerse(j;j⁰) ore_j(j;j⁰) in the above description can be eliminated, (or both of them and a new center introduced, see the proof of theorem II below), whence each of the setsA₁ andA₂is a finite union of cells.

OnA1 the theorem is easily seen to hold, as acf(x;T)ˆF_j(ac(T c(x;j))):

(2:1:12)

OnA2, notice that as the condition

ac(T c(x;j))ˆe_j(j;acf1(x;j);. . .;acfr(x;j))

holds, thus we may follow the steps of Pas' proof on pages (148-154).

The crucial point in these steps is to find a new centerd(x;j) for the cell A₂such that

f(x;T)ˆf⁰(x;d(x;j))(T d(x;j))‡X^d

jˆ2

f^(j)(x;d(x;j))

j! (T d(x;j))^j: This entails that

ordt f(x;T)ˆordtf⁰(x;d(x;j))(T d(x;j))

ˆmin

1jdordt

f^(j)(x;d(x;j))

j! (T d(x;j))^j onA₂, and

acf(x;T)ˆacf⁰(x;d(x;j))ac(T d(x;j));

so,

ordpacf(x;T)ˆordpacf⁰(x;d(x;j))‡ordpac(T d(x;j))

and the second statement of the theorem holds onA2(by eliminating one of the (secondary) centers eor 0, and by observing that ordpacf⁰(x;T) is bounded onA, as acf⁰(x;T)6ˆ0 onA₂and using 2.1.5). p The following statement should be folklore; nevertheless we provide a detailed proof.

LEMMA2.2. Let f_i(iˆ1;. . .;l),be definable functions f_i:C!K((t))

(2:2:1)

where C is a definable subset of K((t))^mKⁿ. Then there exists a partition of K((t))^mKⁿinto definable subsets Xj,such that

acf_i(x;j)ˆg_ij(j;ach1(x;j);. . .;achr(x;j)) (2:2:2)

for all(x;j)2X_j,and where h₁;. . .;h_rare polynomials with integer coef- ficients in x and g_ij is aL_Mac-definable function from a L_Mac-definable subset C⁰K^n‡rinto K,with r is a positive integer.

PROOF. Let us consider first the caselˆ1,f :ˆf1. Asfis a definable function, acf is also definable and its graph is defined by anL-formula

c(x1;. . .;xm;j₁;. . .;j_n;z) in m Val-variables (x1;. . .;xm) and n‡1 RV- variablesj₁;. . .;j_n;z;

c(x1;. . .;xm;j₁;. . .;j_n;z)

acf(x1;. . .;xm;j₁;. . .;j_n)ˆz : (2:2:3†

Incatomic formulas of the formh(x₁;. . .;x_m)ˆ0 (wherehis a polynomial in (x₁;. . .;x_m) with integer coefficients) can be replaced by ach(x1;. . .;xm)ˆ0. We may suppose then that the variables x1;. . .;xm

appear in c only through the RV-terms acfi(x;j) and the Ord1-terms ordth_j(x;j), (iˆ1;. . .;r;jˆ1;. . .;s). Let f be the formula obtained by replacing incacf_i(x;j) by a RV-variabler_i(iˆ1;. . .;r), and ord_th_j(x;j) by a Ord₁-variablel_j, (jˆ1;. . .;s). Then 2.2.3 is equivalent to

(9r)(9l)

"

f(j;z;r₁;. . .;r_r;l₁;. . .;l_s)^ ^^r

iˆ1

acf_i(x;j)ˆr_i

^

^ ^^s

jˆ1

ordthj(x;j)ˆlj

#

Notice thatfdefines the graph of a (partial) function g:KⁿK^rG^s!K (2:2:4)

whose domain is given by Dˆ

(

(j;r₁;. . .;r_r;l1;. . .;ls)2K^n‡rG^sj(9x(x;j)2C) ^^r

iˆ1

acf_i(x;j)ˆr_i

^ ^^s

jˆ1

ord_th_j(x;j)ˆl_j)

The 2-valued-field quantifiers in the above description can be eliminated, whence the domain ofgis definable in the languageLMac.

Let now z;z⁰2K be such that f(j;z;r₁;. . .;r_r;l1;. . .;ls)^

^f(j;z⁰;r₁;. . .;r_r;l1;. . .;ls) holds for some (j;r₁;. . .;r_r;l1;. . .;ls)2D, then we have

9x

c(x;j;z)^c(x;j;z⁰) and sozˆz⁰.

Apply now theorem (1.1) tof, to get

f(j;z;r₁;. . .;r_r;l1;. . .;ls),_^N

jˆ1

x_j^u_j (2:2:5)

where x_j is an LMac-formula, and uj is an LOrd1-formula. We can verify then that x_j defines the graph of a partial function gj:Kⁿk^r!K.

Assume that x_j(j;z;r₁;. . .;r_r) and x_j(j;z⁰;r₁;. . .;r_r) hold, then x_j(j;z;r₁;. . .;r_r)^u_j(l₁;. . .;l_s) and x_j(j;z⁰;r₁;. . .;r_r)^u_j(l₁;. . .;l_s) also hold and then by the above result we should have zˆz⁰.

Finally let X_j ˆ

(x;j)2Cju_j(ord_t(h₁(x;j));. . .;ord_t(h_s(x;j))) ; (2:2:6)

then, for all (x;j)2X_j

acf(x;j)ˆg_j(j₁;. . .;j_n;acf₁(x;j);. . .;acf_r(x;j)):

(2:2:7)

The casel>1 is trivially proved by simultaneous applications of the lemma to each functionf_iseparately and taking intersections.

3. Cell decomposition II.

Analogously to Denef (1986) and Pas (1989), we prove the cell decom- position theorem II, which relies on theorem I.

THEOREM3.1. Let f1(x;T);. . .;fr(x;T)be polynomials as in theorem I, and n12N. Then there exists a finite partition of K((t))^mK((t))in cells.

Each such cell A has parameters(j₁;. . .;j_l)and primary and secondary centers c(x;j)and e(j) respectively such that,for allj2Kⁿ and for all (x;T)2A(j)

ord_t f_i(x;T)ˆord_t(h_i(x;j)(T c(x;j))^mi) and

acf_i(x;T)ˆu_i(ac(T c(x;j));j)ⁿ¹g_i(j)(ac(T c(x;j)) e(j))^yi for iˆ1;. . .;r,and where hi(x;j);gi(j) are definable functions to K((t)) and K respectively;ordpuˆ0andm_i;y_iare non-negative integers that do not depend on(x;j;T).

PROOF. Consider first the caserˆ1;f(x;T):ˆf₁(x;T). In the proof of theorem I, we realize that we can partitionK((t))^mK((t)) in cellsAon which we have

acf(x;T)ˆaca_i0(x;j)ac(T c(x;j))ⁱ⁰

or

acf(x;T)ˆX

i2~I

aca_i(x;j)ac(T c(x;j))ⁱ

where~Iis as in the proof of theorem 2.1. Clearly we need only to consider the case where the cardinality of ~I is greater than 1; consider then the polynomials

F_j(j;j⁰;z)ˆX

i2~I

g⁰_ij(j;j⁰)zⁱ where

g⁰_ij(j;acf₁(x;j);. . .;acf_r(x;j))ˆaca_i(x;j)

for (x;j)2X_j,iˆ12~Iand whereX_j(jˆ1;. . .;s) is a definable subset of K((t))^mKⁿ, and f₁;. . .;f_r are polynomials in x2K((t)) with integer coefficients as in lemma (3.2) above. We can apply theorem II of Denef (1986) separately to each of the polynomialsFj(j;j⁰;z) and then substitute ac(T c(x;j)) forzto get the desired result.

Consider then the caser>1.

Let us now consider the following statementP(A;s):

A is the intersection of s cells,with parameters jˆ(j₁;. . .;j_n) and centers c₁(x;j);. . .;c_s(x;j)and e₁(j);. . .;e_s(j)respectively. Denote by A(j) the intersection of the fibers of the cells of which A is the intersection. For allj,for all(x;T)2A(j),and for iˆ1;. . .;r we have

ord_tf_i(x;T)ˆord_t(h_i(x;j)(T c_h(i)(x;j))^mi) acf_i(x;T)ˆX

k2Ii

ac(a_ki(x;j))ac((T c_h(i)(x;j))^k)

where the hi(x;j)are definable functions,and the non-negative integersm_i, the maph:f1;. . .;rg ! f1;. . .;sgdoes not depend on(x;j;T).

Applying theorem I to each of the polynomialsf₁;. . .;f_rwe get a finite partition ofK((t))^mK((t)) in subsetsAsuch thatP(A;r) holds. The next stepis to show theorem II by descending induction.

Assuming we have a setAand an integers, 1<srsuch thatP(A;s) holds, we should partitionAfurther in a finite number of setsBsuch that P(B;s 1) holds.

Consider two different cells (which have respective centers c₁(x;j),

e1(j) andc2(x;j),e2(j)). By splittingAinto

f(x;T)2A(j)jc1(x;j)ˆc2(x;j)g

and its complement inA, so we will assume thatc₁(x;j)6ˆc₂(x;j). Also, we will assume thate₁(j)6ˆe₂(j) by splittingCinto

f(x;j)2Cje1(j)ˆe2(j)g and its complement inC.

There are four different cases:

(i) ordt(T c1(x;j))>ordt(c2(x;j) c1(x;j)) In this case we have T c₂(x;j)ˆ(T c₁(x;j)) (c₁(x;j) c₂(x;j))

ˆ(c2(x;j) c1(x;j))

1 T c₁(x;j) c1(x;j) c2(x;j)

: As ordt T c(x;j)

c1(x;j) c2(x;j)>0 we get

ordt(T c2(x;j))ˆordt(c2(x;j) c1(x;j)) and

ac(T c₂(x;j))ˆac((c₂(x;j) c₁(x;j)):

(3:1:1) Let

B1ˆ[

j

(x;T)2A(j)jordt(T c1(x;j))>ordt(c2(x;j) c1(x;j))

Then onB1the centerc2is eliminated.

Note that it follows from 3.1.1 that

(ac(T c2(x;j)) e2(j))ˆ(ac( (c2(x;j) c1(x;j)) e2(j))) and thus we can eliminatee₂(j) too.

(ii) ordt(T c1(x;j))<ordt(c2(x;j) c1(x;j)) In this case we have ordt(T c₁(x;j))ˆordt(T c₂(x;j)) and ac(T c₁(x;j))ˆac(T c₂(x;j)).

Let

B₂ˆ[

j

n(x;T)2A(j)jord_t(T c₁(x;j))<ord_t(c₂(x;j) c₁(x;j))o

and we can eliminatec2.

Now note that as ac(T c₁(x;j))ˆac(T c₂(x;j)) we can repeat ex- actly the same arguments in Denef (1986) (page 163) to eliminate one of the centerse₁(j) ande₂(j).

(iii) ordt(T c2(x;j))>ordt(c2(x;j) c1(x;j));

In this case we eliminatec1ande2.

(iv) ord_t(T c₁(x;j))ˆord_t(c₂(x;j) c₁(x;j))ˆord_t(T c₂(x;j)).

In this case we have

ac(c₂(x;j) c₁(x;j))ˆac((T c₁(x;j)) (T c₂(x;j)))

ˆac(T c1(x;j)) ac(T c2(x;j)) where we used the fact that ord_t

1 T c₂(x;j) T c1(x;j)

ˆ0:In this case we can eliminate eitherc1(x;j) or c2(x;j), and we can proceed as before for the elimination ofe1(j) ande2(j).

Now we get a finite partition ofK((t))^mK((t)) in cellsA, such that ordtfi(x;T)ˆordt(hi(x;j)(T c(x;j))^mi)

and

acfi(x;T)ˆX

k2Ii

ac(aki(x;j))bki(j)ac((T c(x;j))^y0ik)

for all (x;T)2Aandm_i;y⁰_ik2N. Finally apply theorem II of Denef (1986) to the polynomials

G_ij(j;j⁰;z)ˆX

k2Ii

g⁰_kij(j;j⁰)b_ki(j)z^y0ik (whereg⁰_kij(j;j⁰) are as in (4.1.6)) to get

Gij(j;j⁰;z)ˆuij(j;j⁰;z)ⁿgij(j;j⁰)(z e(j;j⁰))^yij

(after further partitioning of the cellsA),y_ij2N. It suffices to substitute ac(T c(x;j)) forzin the above to get the result. p

4. Generalized Cell Decomposition.

LetCbe a definable subset ofK^mKⁿG^rS^s.

We call the cells defined abovestrict cells. Now we define a generalized 1-cell (or 1-cell for short) by

Aˆ [

j;z;z⁰

A(j;z;z⁰)

with

A(j;z;z⁰)ˆn

(x;T)2K^mKj (x;j;z;z⁰)2C;ordt(T c(x;j))ˆa(x;j;z);

ord_p(ac(T c(x;j)) e(j))ˆb(x;j;z⁰);(ac(T c(x;j)) e(j))2lP_n1o where jˆ(j₁;. . .;j_n) are variables in the RV sort, zˆ(z₁;. . .;z_r) are variables in the Ord₁ sort,z⁰2S^s are variables in the Ord₂ sort, and the definable functions,c(x;j);e(j) are the centers of the cell, and such that the fibersA(j;z;z⁰) are disjoint for distinct (j;z;z⁰). We remind the reader that Pn1is the set of nonzeron1-powers ofK, wheren1is some positive integer (2) andl2K. The definable setCis called the parameter set of the cellA.

A 0-cell is defined by Aˆ[

j;z;z⁰

A(j;z;z⁰) A(j;z;z⁰)ˆ

(x;T)2K^mKj (x;j;z;z⁰)2C;Tˆc(x;j) for some definable functionc: K^mKⁿ!K.

Note that strict cells falls under the new definition, by adding one Ord1- variable and one Ord2-variable in the following way:

ordt(T c(x;j))ˆz ord_p(ac(T c(x;j))ˆz⁰ and then using the conditions

ordtb1(x;j)}1l1ordt(T c(x;j))}2ordtb2(x;j) and

ord_pd₁(j)p₁l₂ord_p(ac(T c(x;j)) e(j))p₂ord_pd₂(j) to constrain the variableszandz⁰.

Now fix a model (K; K;G;S) for the theory T2, with K ˆK((t)) and GˆSˆZ.

Let us state theorem III (generalized cell decomposition theorem).

THEOREM4.1. Let X be a definable subset of K((t))^mK((t)),and f a definable function from X to K((t)). Then there is a finite partition of X in (generalized) cells A of centers c(x;j)and e(j)such that

ac (f(x;T))ˆg(ac(T c(x;j));x;j) ordt(f(x;T))ˆh(x;z)

ord_p(ac(f(x;T)))ˆh⁰(j;z⁰) where g;h and h⁰are definable functions.

PROOF. Asf andXare definable we have:

(x;T)2Xc(x;T) zˆ(acf)(x;T)c⁰(x;z;T)

zˆord_t(f(x;T))f(x;z;T) z⁰ˆord_p(acf(x;T))f⁰(x;z⁰;T) wherez2K,z2Z,z⁰2Z, andc;c⁰f;f⁰areL formulas.

We assume as usual that the occurrences of (x;T) in c;c⁰;f;f⁰ are uniquely through the RV-terms acf_i(x;T) and the Ord1-terms ordtg_j(x;T), (iˆ1;. . .;r⁰;jˆ1;. . .;s⁰), where f_i andg_j are polynomials in (x;T) with integer coefficients.

Then, applying theorem (1.1), to the conjunctionc^c⁰(for instance) we see that it isT₂-equivalent to

_^q

kˆ1

x_k(z;acf₁(x;T);. . .;acf_r(x;T))^u_k(ord_tg₁(x;T);. . .;ord_tg_s(x;T))

; wherex_k is anL_Mac, andu_kis anL_Ord1-formula.

Applying theorem II to the polynomialsf_i;g_j we can find a finite par- tition of K((t))^m‡1 in cells, each cellAhaving parameter setC and para- meters (j₁;. . .;j_l)2K^l;(z1;. . .;zr)2Z^r, (z⁰₁;. . .;z⁰_s)2Z^s and centers c(x;j) ande(j) such that

acfi(x;T)ˆui(ac(T c(x;j));j)ⁿ¹di(j)(ac(T c(x;j)) e(j))^yi ord_tg_j(x;T)ˆord_th_j(x;j)(T c(x;j))^mj

with ord_pu_iˆ0, for alliˆ1;. . .;r⁰.

It is assumed that each polynomial fi;gj is either identically zero or nowhere vanishing onA(j;z).

We will further partition the cellAon which the theorem holds . OnA(j;z),c^c⁰isT₂-equivalent to

(9r)(9l)(9h)"^^r0

iˆ1

ordph_iˆ0^hⁿ_i¹di(j)(ac(T c(x;j)) e(j))^yiˆr_i

^ _^q

kˆ1

x_k(z;r₁;. . .;r_r0)^u_k(l₁;. . .;l⁰_s)

^ ^^s0

jˆ1

ord_th_j(x;j)(T c(x;j))^mjˆl_j

^u(j;z;ac(T c(x;j));ordt(T c(x;j)))

#

where r₁;. . .;r_r⁰;h₁;. . .;h⁰_r are RV-variables and l1;. . .;l⁰_s are Ord1-vari- ables, anduis the formula that says that (x;T)2A(j;z;z⁰).

We can verify that x_k defines the graph of a partial function g_k :K^r0S^s0 !Kwhose domain is defined by the formula

(9x)(9T)(9j)(9z)

"

uk^u^ ^^r0

iˆ1

ui(ac(T c(x;j));j)ⁿ¹di(j)

(ac(T c(x;j)) e(j))^yiˆr_i

^ ^^s0

jˆ1

ordth_j(x;j)(T c(x;j))^mj ˆl_j# : Call the formula in the bracketsC_k(x;T;j;z;z⁰). Then for all (x;T) such that9j9z9z⁰C_k(x;T;j;z;z⁰) holds, we havezˆg_k(r₁;. . .;r⁰_r):

If for alliˆ1;. . .;r⁰,yiˆ0 and for alljˆ1;. . .;s⁰,m_j ˆ0, no further partitioning of the cell A is required, but we may need to constrain the parameter setCfurther to satisfy the requirements of the theorem.

Foriˆ1;. . .;r⁰such thaty_i6ˆ0 andd_i(j)6ˆ0

(9h_i)ord_ph_iˆ0^hⁿ_i¹d_i(j)(ac(T c(x;j)) e(j))^yi ˆr_i ,(yiordp(ac(T c(x;j)) e(j))ˆordpr_i ordpdi(j))

^(ac(T c(x;j)) e(j))2(r_i=d_i(j))^1=yiPn1; Also,

ord_th_j(x;j)(T c(x;j))^mj ˆl_j

,(m_jordt(T c(x;j))ˆl_j ordth_j(x;j));

forjˆ1;. . .;s⁰.

By theorem 1.1 of (Scowcroft & van den Dries 1988) there exists a partition of theLMac-definable set Proj_KⁿCintoLMac-definable subsetsD, on each of which the definable function d_i(j) is analytic. Thus, by parti- tioning the definable sets D further if necessary we can assume that (r_i=d_i(j))^1=yi have constantn₁-th power residue, hence (r_i=d_i(j))^1=yiP_n1ˆ

ˆl_iP_n1 for somel_i2K.

In the above description we notify the reader that our cell decomposi- tion may contain 0-cells (if we havel_jˆ 1,m_j 6ˆ0 and ordth_j(x;j)<1for somej).

Also, using the observation that given two n^thpower residues having non-empty intersection, one of them must contain the other, we deduce that ac(T c(x;j)) e(j)2liPn1for somei.

Finally notice that a function given conjunctly by definable conditions is a function given by the conjunction of these conditions, hence the result follows.

The remaining statements of the theorem are left to the reader. p

REFERENCES

[1] R. CLUCKERS - F. LOESER, Constructible motivic functions and motivic integration, preprint, math. arxiv 2004.

[2] J. DENEF, The rationality of the Poincare series associated to the p-adic points on a variety, Invent. Math.77(1984), pp. 1-23.

[3] J. DENEF,p-adic semi-algebraic sets and cell decomposition, J. reine angew.

Math.369(1986), pp. 154-166.

[4] I. FESENKO, Measure,integration and elements of harmonic analysis on generalized loop spaces, www.maths.nott.ac.uk/personal/ibf/aoh.pdf, 2003.

[5] E. HRUSHOVSKI - D. KAZHDAN, Integration in valued fields, preprint, math.arxiv 2005.

[6] J. IGUSA, An introduction to the theory of local zeta functions, AMS/IP Studies in Advanced Mathematics, 14. International Press, Cambridge, MA, 2000.

[7] A. MACINTYREOn definable subsets of p-adic fields, J. Symb. Logic41(1976), pp. 605-610.

[8] J. PAS,Uniform p-adic cell decomposition and local zeta functions, J. Reine Angew. math.,399(1989), pp. 137-172.

[9] P. SCOWCROFT- L.VAN DENDRIES,On the structure of semialgebraic sets over p-adic fields, J. Symbolic Logic,53(1988), pp. 1138-1164.

Manoscritto pervenuto in redazione il 12 settembre 2005