The divisor classes of the surface \(z^ p=G(x,y)\), a programmable problem
The twin fields of research, computational algebraic geometry and computational number theory, have during the last ten years been the scene of an increasing and lively research activity. Drawing from the enormous wealth of mathematical substance accumulated in two of the oldest mathematical disciplines, number theory and algebraic geometry, on the one hand, and the vast open possibilities of the newest of our sciences, computer science, on the other, this should not be surprising. - The mutual enrichment of the two sides of this dialogue can hardly be overemphasized: Techniques from algebraic geometry and number theory can for instance yield new information in complexity theory or in coding theory, while the computational tools developed and under development in computer science, such as symbolic manipulation and computer algebra, parallel processing or more advanced ideas from research in artificial intelligence, carry a great potential for the advancement of mathematics in general and algebraic geometry and number theory in particular. On a more modest level one frequently finds that rather simple mathematical techniques can acquire new and increased power when utilized in a modern computational setting. Even if no ``new theorems'' are proved, new information is gained in several ways: Computations of explicit examples may show patterns, and pave the road for future theoretical advances. The computational techniques employed may carry over to similar situations, and thus be of general significance even if the work in question is of a rather special nature. The paper under review falls into this last category. Let k be an algebraically closed field of characteristic \(p>0.\) Let \(F\subset {\mathbb{A}}^ 3_ k\) be an affine surface defined by an equation of the form \(z^ p=G(x,y)\) where \(G(x,y)\in k[x,y]\). The article develops an algorithm for computing the divisor class group \(Cl(F)\) of F. It is assumed that \(\partial G/\partial x\) and \(\partial G/\partial y\) have no common factors. We put \(g=\deg(G)\). The algorithm is derived from the two theorems below, shown by \textit{J. Lang} in J. Algebra 84, 371-410 (1983; Zbl 0528.14017): Theorem 1: The surface F is normal, and \(Cl(F)\) is isomorphic to the additive group of polynomial solutions \(t(x,y)\in k[x,y]\) of degree \(t<g- 2\) of the following system of equations \(\nabla (G^ jt)=0\), \(j=0,1,...,p-2\), \(\nabla (G^{p-1}t)=t^ p\), where \(\nabla =\partial^{2p-2}/\partial x^{p-1}\partial y^{p-1}.\) Theorem 2: \(Cl(F)\) is a finite p-group of type \((p,p,...,p)\) of order \(p^ M\), where \(M\leq g(g-1)/2.\) Using this, one associates to each surface a system of p-linear equations with linear equations and equations of the form \(a_ 1x_ 1+...+a_ nx_ n=x^ p_ j\), where the \(a_ i\) belong to the field \(F_ G\) obtained by adjoining the coefficients of G to the prime field of k. The algorithm, given implicitly through a series of lemmas, uses elementary techniques like determinants and row deduction similar to those from linear algebra. - As a consequence, the authors are also able to calculate quickly the order of \(Cl(F)\) in many cases. It is asserted that the problem of explicitly computing the set of solutions itself ``does not seem to be programmable for the time being''. The reason given for this is that ``we do not know in what and how large an extension of the field \(F_ G\) the solutions lie''. To find the required extension of \(F_ G\) referred to above is an interesting problem. It would seem that a well-written algorithm for the problem treated in the article under review should return these data, and a computer program utilizing the most recent tools from computer algebra should be able to carry out the computations in practice, at least in some cases. Unfortunately the algorithm is not explicitly given in the article, and no information is given concerning the computer program and the computations themselves. Some formulations in the paper make it look as if though more details were included in a previous version, they have been deleted - perhaps at the request of a referee? If this is so, it would be a real pity, since the information carried by the rather interesting paper under review is significally reduced by the absence of this information.
algorithm for computing the divisor class group
scientific article; zbMATH DE number 4019198
scientific article