数理解析研究所講究録
第 1708 巻 2010 年 26-28

26

A decomposition theorem in$K_{ex}$Masanori Itai
Deapartemnt of Mathematical Sciences
Tokai University, Hiratsuka, Japan*Abstract
and $E=\{y_{1}-e^{x_{1}}, \cdots, y_{n}-e^{x_{n}}\}$ . A theorem ofLet bc a finitc subset ofRichardson gives all irreducible $deco|||positiol|$ of tlie set $defi_{l1}ed$ a the zero set of and . The same
, an algebraically closed field of characteristic zero with pseudo-exponentiation.
statement holds on$S$$\mathbb{Q}[x_{1}, \cdots, x_{n}, y_{1}, \cdots , y_{n}]$

$E$

$S$

$s$

$K_{ex}$1

Decomposition of exponential-algebraic sets

1.1

Richardson‘s Theorem in$\mathbb{C}_{ex}$In this sub section, we work in
the field of’ complex numbers with the complex expoential.
, . Suppose,
and $E=\{y_{1}-$Let.
and$C^{2n}(S,E)=\{(x,y)p_{i}(x,y)=0,y_{j}-e^{x_{j}}=0,(i=1,\cdots,
k,j=1,\cdots, n)\}$ .
,. PutIn [Ri], Richardson proves the following theorem.
$\mathbb{C}_{cxp}$.

$\overline{x}=x_{1},$$e^{x_{1}},$$\cdots$$y,$も$-e^{x\prime}$$\cdots$$x_{n}$$\overline{y}=y_{1},$$\cdots$$S=\{p_{1}(\overline{x},\overline{y}), \cdots,p_{k}(\overline{x},\overline{y})\}$$y_{\mathfrak{n}}$$\}$Theorem 1 (Richardson) For any finite set
be written as a union of finitely many sets;$S\subset \mathbb{Q}[\overline{x},\overline{y}]$, the zero set of exponential system $(S, E)$ can$\mathbb{C}^{2n}(S, E)=\bigcup_{:}C_{i}$where each set$C_{i}$

is the zero set of a triangular condition$\Delta_{i}$.

To makc thc $stat_{C1}iient$ precise we need some definitions.
First introduce a term-order among variables, e.g.,.
a well-founded order on the polynomials in
Let be a finite set of polynomials$x_{1}\prec x_{2}\prec\cdots\prec x_{n}$. With this order we introduce$\mathbb{Q}[x_{1}, \cdots, x_{n}]$, we say that is an. Ifascending set.
define t,he notion of triangular condit,ion we need t,he following definition of a diffe. $re$ nlial malnx.
$S$$\{p_{1}(\overline{x},\overline{y}), \cdots,p_{k}(\overline{x},\overline{y})\}$$S$$p_{1}\prec\cdots\prec p_{k}$$\prime I^{\backslash }o$Definition 2 Let$S$

be a finite subset of$df(S, E_{1})=( \frac{\partial}{\partial}R\frac{\partial}{\partial}R1000x_{1}x^{\frac{k}{1}}$.

$-y_{0}00^{\cdot}$$\mathbb{Q}[x, y]$and$E_{1}=\{y_{i_{0}}-e^{x_{i_{0}}}, \cdots, y_{i_{l}}-e^{x_{t_{\iota}}} \}\subseteq E$$\#_{0}^{\partial_{x}}\frac{k}{n}\overline{\partial}x_{n}^{L}\partial x_{0}0^{\cdot}$.

$\star_{o^{1}}^{\partial_{V^{k}}^{y_{1}}}\frac{\partial}{\partial}2100^{\cdot}$$\star_{0’}^{\backslash }\overline{\partial}y_{k}\partial_{J\mathfrak{n}}\delta_{1L}00^{\cdot})$.

$00^{\cdot}$$-y_{l}00^{\cdot}.\cdot$$001^{\cdot}$.

$00^{\cdot}$$001^{\cdot}$

Deflnition 3 (Triangular conditions) $(S, E_{1})$ is a triangular condition, if
1.
is an ascending set.
2. Let be the determinant of a maximal minor of the dimatrix $df(SmE_{1})$ . Noticethat is$Rem(J,S)\neq0$.
polynomial.
Weinferthatawhich $(S=0,$ $E_{1}=$is such that if $C(t)$ is any smooth curve in
3. Let $E_{2}=E-E_{1}$ . Then$I\neq 0,$ $J\neq 0)$ , then any $y_{i}-e^{x:}\in E_{2}$ is either identically zero on $C(t)$ or nevee zero.
$S$$J$$J$$fferenti_{C}\backslash 1$$\mathbb{C}^{2\mathfrak{n}}$$E_{2}$$0,$‘This work was supported by Grant-in-Aid for. IkOsearch (C) 19540146.

$S(,i(^{\backslash }.nti\Gamma)(^{\backslash }$27
Decomposition of $(S, E)=0$ in
Since the proof of Theorem 1 uses only algebraic tools, the same statement holds in

1.2$K_{ex}$Theorem 4 For any finite set
union of finitely many sets;$S\subset \mathbb{Q}[\overline{x},\overline{y}]$$K_{cx}$.

, the zero set of exponential system $(S, E)$ can be written as a$K^{2n}(S, E)= \bigcup_{i}C_{i}$Proof: Given a set.
onto

$S$

, we just proceed the decomposition of$\mathbb{C}^{2n}(S, E)$. Then transfer the decomposition$K^{2n}$1.3$\blacksquare$

Exisitential closedness and the decomposition

Definition 5 Let $(m_{1j})\in M_{n}(Z)$ be an
Then-matrix. Consider a tuple of variables
denotes the following tuple:$(u_{1}, \cdots, u_{l}, v_{1}, \cdots , v_{n})^{(\pi l:g)}$$( \sum_{j=1}^{n}m_{1j}u_{j},$We call$(\overline{u}, \overline{v})^{(m}j$$n\cross n$$(u_{1}, \cdots, u_{n}, v_{1}, \cdots,v_{\mathfrak{n}})$.

$\cdots,\sum_{j=1}^{n}m_{nj}u_{j},\prod_{j=1}^{n}v_{j}^{m_{1j}},$ $\cdots,\prod_{j=1}^{n}v_{j}^{m_{nj}})$an admissible transformation of the tuple of variables (ti, ).
$\overline{v}$Recall that the strucutre$K_{cx}$satisfies following the existential closedness condit,ion :Existential closednessLetbe an irreducible, ,of $po$ lynomials with coefficints and,,its generic zero. If the system
satisfies the following conditions called normality
and freeness then there is a generic zero ofsuch that$P_{\overline{a}}(x_{1}, \cdots x_{n},y_{1}, \cdots , y_{n})$$s^{\backslash }yste1\gamma i$$y_{\tau\iota}^{0})\in K^{2n}$$y_{1}^{0}$$(x_{1}^{0},$$\overline{a}$$\cdots$$x_{n}^{0}$$P_{\overline{a}}(\overline{x},\overline{y})$$P_{\overline{a}}$.
.

$y_{i}^{0}=ex(x_{i}^{0})$(Normality condition) For any distinct$i_{1},$,$\cdots,i_{m}$we havetr.

$i=1,$ $\cdots,$ $n$ .

after any admissible transformation of variables,$\deg_{Q(\overline{a})}(x^{0_{1}}, \cdots x_{i_{m}}^{0}, y_{i_{1}}^{0}, \cdots, y_{i_{n\iota}}^{0})\geq m$(Freeness condition) After any admissible transformation of variables, we have for all$x_{i}^{0}\not\in$acl$(\mathbb{Q}(\overline{a}))$and$y_{i}^{0}\not\in$acl$i$$(\mathbb{Q}(\overline{a}))$So, the exisitential closedness assures the existence of generic solutions to the exponential system$(P, E)$ if $P$ is normal \‘and free.

On the other hand, the decomposition theorem describes the structure of the solution set to theexponential system if it has solutions.

1.4

PQF is not enough for an analytic Zariski structure

PQl’ stands for the positive quantifier free topology, i.e., the poitive quantifier free definable sets forming
thefor a topology. Since the set of positive quantifier free definable sets is closed under finite unions
and finit,
PQF
be.
inSuppose we topologize
with PQF. Then the decomposition theorem above (Theorem 4) suggeststhat the irreducible subsets ofdefined as the zero sets of cxponential systeln $ca$11 be viewed as analytic
set.
However as Zilber remarks we see that PQF is not rich enough forbeing an analytic Zariski
structure :First notice that the properties ofare all of algebraic nature; $K$ being
closed field,
ex being a homomorphism from t,he additive struct,ure to $th_{P}m\iota\iota 1|,ip(’ r11ive$ structure 0 the field $K$ andthe existential closedness guaranteeing the existence of solutions to exponential system.
Therefore in order to consider
as an \‘analytic Zariski structure, we need to say that e.g., ex’ ex.
For this, we need to define
derivative of ex should be definable and analytic.
$b_{\epsilon}\backslash sis$$eint,ert^{\backslash },\epsilon r’ 1.ion\backslash ^{\backslash },$$\downarrow_{i}he$$t_{f}m$$in\downarrow.rod_{11(}\cdot e,(1$$K_{cx}$$K_{cx}$$K^{2n}$$K_{ex}$$K_{ex}$$algebr_{C}’\iota ic_{C}\sqrt 1y$$1^{\cdot}$$K_{cx}$$t\}_{1}e$$=$28
Consider first$E=\{(z, x_{1}, x_{2}):z(x_{1}-x_{2})=ex(x_{1})-ex(x_{2})\}$

Then clearly $\dim E=2$ . Next put $E’=E\cap\{(z, x_{1}, x_{2}) : x_{1}=x_{2}\}$ . We see also that $\dim E’=2$ . Since$E’$ is a subset ofof equal dimenstion, is not irreducible if $E$ is analytic. Therefore, by an axiom of.
analytic Zariski structure we have that $E=E^{0}\cup E’$ for some analytic set
Consider the graph of the following function
$E$

$E$

$E^{0}$$g(x_{1},x_{2}):=\{\begin{array}{ll}\frac{ex(x_{1})-ex(x_{2})}{x_{1}-x_{2}} x_{1}\neq x_{2}ex (x_{1}) otherwise\end{array}$

Notioe here that$E^{0}\backslash E’$is the graph of$g(x_{1}, x_{2})= \frac{ex(x_{1})-ex(x_{2})}{x_{1}-ex_{2}}$,$x_{1}\neq x_{2}$is the graph of the derivative of ex.
and thatis quantifier free$g(x_{1},x_{2})$ above should be Zariski continuous. Although the graph ofThe function
definable, it is not a closed set in the PQF.
being an analytic Zariski. For
This simple example shows that the PQF is not rich enough fordetails see p. 189 of [Z3].
$E^{0}\cap E’$$g$$K_{cx}$

References[CLO] D. Cox, J. Little and D. 0‘Shea, Ideals, Varieties, and Algorithms, Springer, 1992
[HZ] Ehud Hrushovski, Boris Zilber, Zariski geometries, J of the AMS, 1996
Logicfor Mathanalytic Zariski structure over a field,
NicJc Peatfi[Pl][P2] Ni Peatfield, A complex-like Analytic Zariski structures on models 01’ t,he theory of a genericfunction with derivatives[PZ] Ni(’k Peatfield, Boris Zilber, Analytic Zariski $strtl(’ t\iota lrPS$ and the Hrushovski construction, Annalsof Pure and Applied Logic 132(2005) 127-180
D. Richardson, A Zero structure Theorem for Exponential Polynomials, Proccedings of the 1993
[Ri]
international symposium on Symbolic and algebraic computation, 144-151, 1993
[Z3]
Boris Zilber, Zariski Geometries, 20 M\‘arch 2009 version$(^{\backslash ]d},$$(^{\backslash }k$$\Lambda n$$\Lambda rt’ h]_{(}\backslash$$(.m_{f}\backslash tit’ a1$