Model completeness results for expansions of the ordered field of real numbers by restricted Pfaffian functions and the exponential function
Model completeness results for expansions of the ordered field of real numbers by restricted Pfaffian functions and the exponential function
复制标题
DOI：
10.1090/s0894-0347-96-00216-0
复制
发表时间：
1996
影响因子：
3.9
通讯作者：
A. Wilkie
中科院分区：
文献类型：
--
作者：
A. Wilkie
回想一下，如果R的子集可以表示为形式为{~α∈R：p(~α)=0}，{~α∈R：q(~α)>；0}的集合的(有限)布尔组合，则称其为半代数，其中p(~x)，q(~x)是具有实系数的n元多项式。一个从R到R的映射称为半代数，如果它的图被认为是R的子集，则称它是半代数的。这种集合和映射的几何(“半代数几何”)现在是一个被广泛研究和蓬勃发展的学科，这在很大程度上要归功于逻辑学家阿尔弗雷德·塔尔斯基在20世纪30年代的基础性工作。他证明了([11])半代数集在半代数映射下的像是半代数的。(一个熟悉的简单实例：投影图R×R∈R下的{<a，b，c，x>→R：a6=0和ax+bx+c=0}是{<a，b，c>∈R：a 6=0和b−4ac≥0}。)塔斯基的结果表明，这类半代数集在一阶逻辑可定义性下是闭的(其中，以及布尔运算，量词“∃x∈R.。。“和“∀x∈R.。。“是允许的)，并且出于这个原因，它被逻辑学家称为“R上有序环结构的量词消除”。直接结果是半代数集的闭包、内部和边界是半代数的事实。它也是半代数几何中许多归纳论点的基础，其中给定的半代数集的期望性质是从该集到低维的投影的相同性质中推断出来的。例如，用这种方法证明了(由于Hironaka)任何有界半代数集都可以三角化的事实。在20世纪60年代，分析几何学家Lojasiewicz将上述理论扩展到分析背景下([8])。R的半解析子集的定义与上面相同，只是对于基本集，允许p(~x)‘S和q(~x)’S是解析函数，并且我们只坚持布尔表示在R的每个点附近局部工作(允许在不同点附近有不同的表示)。还需要限制映射是适当的(具有半解析图)。在这种限制下，如果目标空间是R或R，则半解析集的象是半解析的，这是真的。从本世纪初开始，m≥3的映射到R的反例就已为人所知。(它们是由于奥斯古德，见[8]。)然而，1968年Gabrielov([5])澄清了这种情况，他证明了次解析集类
Recall that a subset of R is called semi-algebraic if it can be represented as a (finite) boolean combination of sets of the form {~ α ∈ R : p(~ α) = 0}, {~ α ∈ R : q(~ α) > 0} where p(~x), q(~x) are n-variable polynomials with real coefficients. A map from R to R is called semi-algebraic if its graph, considered as a subset of R, is so. The geometry of such sets and maps (“semi-algebraic geometry”) is now a widely studied and flourishing subject that owes much to the foundational work in the 1930s of the logician Alfred Tarski. He proved ([11]) that the image of a semi-algebraic set under a semi-algebraic map is semi-algebraic. (A familiar simple instance: the image of {〈a, b, c, x〉 ∈ R : a 6= 0 and ax +bx+c = 0} under the projection map R×R→ R is {〈a, b, c〉 ∈ R : a 6= 0 and b−4ac ≥ 0}.) Tarski’s result implies that the class of semi-algebraic sets is closed under firstorder logical definability (where, as well as boolean operations, the quantifiers “∃x ∈ R . . . ” and “∀x ∈ R . . . ” are allowed) and for this reason it is known to logicians as “quantifier elimination for the ordered ring structure on R”. Immediate consequences are the facts that the closure, interior and boundary of a semialgebraic set are semi-algebraic. It is also the basis for many inductive arguments in semi-algebraic geometry where a desired property of a given semi-algebraic set is inferred from the same property of projections of the set into lower dimensions. For example, the fact (due to Hironaka) that any bounded semi-algebraic set can be triangulated is proved this way. In the 1960s the analytic geometer Lojasiewicz extended the above theory to the analytic context ([8]). The definition of a semi-analytic subset of R is the same as above except that for the basic sets the p(~x)’s and q(~x)’s are allowed to be analytic functions and we only insist that the boolean representations work locally around each point of R (allowing different representations around different points). It is also necessary to restrict the maps to be proper (with semi-analytic graph). With this restriction it is true that the image of a semi-analytic set, known as a sub-analytic set, is semi-analytic provided that the target space is either R or R. Counterexamples have been known since the beginning of this century for maps to R for m ≥ 3. (They are due to Osgood, see [8].) However, the situation was clarified in 1968 by Gabrielov ([5]) who showed that the class of sub-analytic sets