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- W2093042371 abstract "The notion of mathematical crystal growth was used by D. A. Klarner [Kl, K2] to study spreading of information in a large computer net. His main result states that the generating function of a growing crystal is always a rational function with explicitly determined denominator. In a quite different context, W. Scharlau [S] proved that the generating function of the Witt ring of a Pythagorean or SAP field (with finitely many square classes) is also a rational function with a unique pole. He conjectured that the result holds for every finitely generated Witt ring. We extend Klarner’s result to include the more general class of weighted crystals and then we show that the generalization is strong enough to prove Scharlau’s conjecture. Let N be the set of non-negative integers and, for r > 0, let f: Z' + N be a function with finite support supp f = (g E Z’lf(~) # 01. Suppose a finite set of particles is located at points of Z’ in r-dimensional Euclidean space, each particle having its own “weight.” We agree that the particle at tl has weight f(a) (and iff(a) = 0, then there is no particle at the point a). Thus our function f describes completely the finite set of particles and we will call f a weighted crystal. If all particles have weight 1, then we say that the crystal is simple. By a growing simple crystal we mean a sequence of simple crystals (fnl y1 E N} such that there exists a finite subset D of E’ such that suppf,+,=D+suppf,= (v+alv~D, a~suppf~} for every nEN). The set D will be called an increment set of the crystal. A growing weighted crystal is a sequence of weighted crystals, which can be represented as a sum of growing simple crystals. More precisely, {f, (n E N} is a growing weighted crystal if there are growing simple crystals (ft’: n E N }, i = 1, . . . . k, such thatf,=fy)+ ... +fP) for every rz~ N. 190 0021-8693191 $3.00" @default.
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- W2093042371 date "1991-01-01" @default.
- W2093042371 modified "2023-09-29" @default.
- W2093042371 title "Crystal growth and Witt rings" @default.
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- W2093042371 doi "https://doi.org/10.1016/0021-8693(91)90074-i" @default.
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