The Brianchon-Gram and Sommerville theorems on angle-sums for convex polytopes and polyhedral cones are here shown to be particular cases of an angle-sum relation for general polyhedral sets. The new relation is proved on the level of an equidissectability theorem, and this approach yields yet other angle-sum relations, including a different generalization of the Brianchon-Gram theorem. Further results extend, again to equidissections, earlier angle-sum relations of the author and others.

In [1] I made the following conjecture. If we tile a convex polygon of at most six sides with N convex tiles of areas a1, …,aN, then the total perimeter of the tiles is never less than the total perimeter of N regular hexagons of areas a1, …, aN. In order to show that the condition of the convexity of the tiles cannot be omitted I constructed a tiling with an equal number of “pentagons” and “heptagons” of perimeters and areas and , and ā5 and ā7, respectively. In a letter to me R. Schneider kindly pointed out that, in contrast to the value given in [1], the value of the quotient was equal to 3.7263… This being greater than √8√3 = 3–7224… the tiling given in [1] does not yield the desired counter-example. In what follows we shall construct a tiling for which the respective quotient will turn out to be less than √8√3.

A (convex) d-polytope is the convex hull of a finite set of points in Euclidean d–space Ed. The (Minkowski) sum of two polytopes P1 and P2 is defined by

.

Since Minkowski [29] gave his famous lattice point theorem for centrally symmetric convex bodies, a theorem that turned out to be of fundamental importance in number theory, much effort has been made to obtain tight estimates for the number of lattice points of a given lattice in convex bodies in terms of the basic quermass-integrals Wo,…, Wd, whose eminent role shows in Hadwiger's functional theorem [14, 15, 16, see also 17, p. 221–225]. (For the discrete analogues of Wo,…, Wd see [2].) The present paper is concerned with an upper estimate of this kind.

This paper is devoted to the establishment of explicit bounds on the rational function solutions of a general class of equations in several variables. The first general result on Diophantine equations over function fields was discovered in 1930 [1], as an analogy on the work of Thue on number fields: it was shown that the degrees of polynomial solutions X, Y in k[z] of f(X, Y) = c are bound for each c: here f denotes an irreducible binary form over k[z] of degree at least three, and k is an algebraically closed field of characteristic zero. The Manin-Grauert theorem [7] extended this conclusion to the rational function solutions X, Y in k(z) of any equation in two variables over k(z), provided that the curve corresponding to the equation has genus two or more: this is the analogue for function fields of the Mordell conjecture for number fields, proved by Faltings in 1983. Parsin [6] made the Manin-Grauert theorem effective by furnishing explicit bounds on the degrees of X and Y: this approach followed Grauert and Shafarevitch in relying heavily on algebraic geometry. In 1976 a different attack was made by Schmidt [8] using the theory of algebraic differential equations, first developed by Kolchin and Osgood. This method produced very good bounds for equations such as the Thue equation discussed above: for example, Schmidt provides the bound

for any rational function solution X, Y in k(z) of f(X, Y) = 1, where f denotes a binary form over k(z) of degree in X and Y at least five, and which factorises into distinct linear factors. Here deg zf is the degree of f in z. In 1983 [2] the power of Diophantine approximation provided another approach to attack Dlophantine equations.over function fields. By first establishing an inequality on solutions of the unit equation in two variables, it was shown in [3, p. 122] that the bound (1) could be improved to 36 degz f, and the condition on the degree in X and Y could be relaxed to include quartic forms. In fact the principal purpose of this new approach using Diophantine approximation had been to provide algorithms for the effective construction of all the polynomial solutions of various equations: it was a surprise to find that not only were these algorithms extremely efficient, but also that the technique extended to include rational function solutions. A full account of this approach and its consequences is given in a recent tome [3].

Let 0 < γ ≤ γ” denote the ordinates of consecutive nontrivial zeros of ζ(s) and set

In this paper we give alternative proofs of some formulas of Shintani [1, 2] and Zagier [3] concerning Kronecker limit formulas for real quadratic fields.

Let K be an algebraic number field of degree n = r1 + 2r2 (in the usual notation) over the rationals. ZK will denote the ring of integers in K. Consider the set of all totally positive primitive roots modulo the square of a prime ideal p of first degree in K. We recall (see e.g., [6], p. 249) that there exists such a primitive root mod p2, if, and only if, p is of first degree. Let vp be a least element of this set, least in the sense that its norm Nvp is minimal. We ask for the order of magnitude of Nvp in comparison to Np2. The author”s work [5] on cubefree ideal modulus character sums yields the estimate

for any a > 0, where the implied «-constant depends on a and K.

The purpose of this paper is to define and study a natural rank function which associates to each differentiable function (say on the interval [0,1]) a countable ordinal number, which measures the complexity of its derivative. Functions with continuous derivatives have the smallest possible rank 1, a function like x2 sin (x−1) has rank 2, etc., and we show that functions of any given countable ordinal rank exist. This exhibits an underlying hierarchical structure of the class of differentiable functions, consisting of ω1, distinct levels. The definition of rank is invariant under addition of constants, and so it naturally assigns also to every derivative a unique rank, and an associated hierarchy for the class of all derivatives.

Throughout this paper A is a commutative noetherian ring (with identity) and M is an A-module. We use to denote, for i ≥ 0, the i-th right derived functor of the local cohomology functor L with respect to an ideal a of A [8; 2.1].

Throughout this note the letters D and K denote a commutative integral domain with 1 and its field of fractions.