Exponential formula

Abstract: 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

Abstract: LetU=(U(t, s)) t≥s≥O be an evolution family on the half-line of bounded linear operators on a Banach spaceX. We introduce operatorsG O,G X andI X on certain spaces ofX-valued continuous functions connected with the integral equation $$u(t) = U(t,s)u(s) + \int_s^t {U(t,\xi )f(\xi )d\xi }$$ , and we characterize exponential stability, exponential expansiveness and exponential dichotomy ofU by properties ofG O,G X andI X , respectively. This extends related results known for finite dimensional spaces and for evolution families on the whole line, respectively.

Abstract: The basic objects of this study are exponential sums on a variety defined over a finite field Fq (q = pa, p = char Fq). As we have remarked in some earlier articles [1], [2], we find it more natural to begin with exponential sums on the torus (Gm)n, extend via the usual toric decomposition of An to exponential sums on affine n-space, and finally proceed via a standard character argument [4] to exponential sums on an affine variety defined over Fq. While this is the natural order of the work, what we do, in fact, in the first part of this article is to combine the first two steps and deal with exponential sums on varieties V of the form V = (Gm)r X As (r + s = n). Let f E Fq[xl,..., xn, (xl ... xr-'] be an arbitrary regular function on V. Then f is a sum of monomials and as such has a well-defined Newton polyhedron A( ff) at infinity. This is the convex closure in Rn of the lattice points which occur as exponents of the terms of f together with the origin. We have indicated in our previous work [1], [2] the description of some of the invariants of the associated L-function in terms of properties of this polyhedron. For example, in [1] we showed how bounds for the degree and total degree of the L-function associated with a general exponential sum on V can be expressed in terms of the volumes of A(f) and the intersections of A(f) and the various coordinate spaces. In the present article, assuming f is nondegenerate and commode with respect to Af f), we show these estimates are sharp. Our methods are p-adic and are based on the work of Dwork [11], [12]. Our main accomplishment, from which our other results follow, is the extension of Dwork's cohomology theory from smooth, projective hypersurfaces in characteristic p to a general class of exponential sums. Given f regular on V, we construct a complex of p-adic Banach spaces on which Frobenius acts. The alternating product of characteristic polynomials of Frobenius describes the associated L-function. In fact, when f is nondegenerate and commode with respect to A( f), the complex is acyclic in dimensions other than 0 and the characteristic