A. Seidenberg

The ritual origin of geometry

SummaryLet us sum up the history of geometry from its beginnings in peg-and-cord constructions for circles and squares.The circle and square were sacred figures and were studied by the priests for the same reason they studied the stars, namely, to know their gods better. The observation that the square on the diagonal of a rectangle was the sum of the squares on the sides found an immediate ritual application. Its elaboration in the sacrificial ritual gave it a dominant position in ancient thought and ensured its conservation for thousands of years. This initial elaboration took place well before 2000 B.C. By 2000 B.C., it was already old and had diffused parts of itself into Egypt and Babylonia (unless, indeed, one of these places was the homeland of the elaboration). These parts became the basis of a new development in these centers. The new, big development was the solution of the quadratic. A thousand years and more later, Greece inherited algebra from Babylonia, but its geometry has more of an Indian than a Babylonian look. It inherited geometric algebra, the problem of squaring the circle, the problem of expressing √2 rationally, and some notions of proof.

Construction of the integral closure of a finite integral domain

SuntoIl problema qui trattato riguarda la costruzione della chiusura intera di un anello intero finitok[x1,x2,..,xn]. Se la caratteristicap è zero risulta possibile effettuare la suddetta costruzione per ogni campok dato in modo esplicito; sep > 0 occorre assegnare una condizione aggiuntiva.SummaryThe problem is to construct the integral closure of a finite integral domaink[x1,..,xn]. If the characteristicp is zero, this can be done for any explicitly given fieldk; ifp > 0, another condition must be imposed.

On the Volume of a Sphere

1. Archimedes on the volume of a sphere It is well known that Archi edes found the volume of a sphere. If S is the volume of a sphere of radius a, then S = y Tia3, where tz/4 is a constant giving the ratio of the area a circle to the area of its circu scribed square. Hence, too, volume (= n 2 • Id) of th right ci cular cylinder of h ight 2a circumscribing the sphere is y S. Archimedes expressed the wish that the figure of a sphe e a d a circumscribing cylinder be ng ave on his tombston (cf. Ball 1901, p. 69). For us, finding the volume is a e sy exercise. If the center of the sphere is taken at the origin of a rectangular coordinate system, the equation of the sphere is x2 + y2 + z2 = a2. The section of this by a plane at level z is a circle of radius 1 a2 z2. Hence the volume, as taught in our first y ar calculus c urses, is a

On analytically equivalent ideals

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On the functions invariant under the action of an algebraic group on an algebraic variety

SuntoSiaG un gruppo algebrico con componentiG1, …,G3,V una varietà, e siaG operante regolarmente suV. Si domanda come si trovano tutte le funzioni razionali suV invarianti sotto l’azione diG suV. Siak un campo di definizione diG, diV, e delle azioni diG suV. SianoV un punto generico diV/k, Gv l’orbita div, eO (v) la chiusura diGv. Allora si definiscono la forma e la forma ridottaF di Chow d’O (v). E’ provato che i quozienti dei coefficienti diF generano soprak il campo delle funzioni razionali invarianti definite soprak contenuto ink(v).SummaryLetG be an algebraic group having componentsG1, …,G3,V a variety, and letG operate regularly onV. The problem is to find all the rational functions onV invariant under the action ofG onV. Letk be a field of definition ofG, ofV, and of the action ofG onV. Letv be a generic point ofV/k,Gv the orbit ofv, andO (v) the closure ofGv. One defines the Chow form and reduced Chow formF ofO (v). It is proved that the quotients of the coefficients ofF generate overk the full field of invariant rational functions defined overk contained ink(v).

Existence of a non-singular curve of order 7 and genus 4 in 3-space

SuntoSi mostra come si costruisce una curva del tipo descritto nel titolo.SummaryIt is shown how to construct a curve of the kind described in the title.