Abstract

Alt’s problem, formulated in 1923, is to count the number of four-bar linkages whose coupler curve interpolates nine general points in the plane. This problem can be phrased as counting the number of solutions to a system of polynomial equations which was first solved numerically using homotopy continuation by Wampler, Morgan, and Sommese in 1992. Since there is still not a proof that all solutions were obtained, we consider upper bounds for Alt’s problem by counting the number of solutions outside of the base locus to a system arising as the general linear combination of polynomials. In particular, we derive effective symbolic and numeric methods for studying such systems using probabilistic saturations that can be employed using both finite fields and floating-point computations. We give bounds on the size of finite field required to achieve a desired level of certainty. These methods can also be applied to many other problems where similar systems arise such as computing the volumes of Newton-Okounkov bodies and computing intersection theoretic invariants including Euler characteristics, Chern classes, and Segre classes.

摘要

Alt 于 1923 年提出的问题是计算四连杆机构的数量，其耦合器曲线在平面上插入九个一般点。这个问题可以表述为计算多项式方程组的解的数量，该方程组首先由 Wampler、Morgan 和 Sommese 在 1992 年使用同伦延拓进行数值求解。由于仍然没有证据表明所有解都已获得，因此我们认为通过计算作为多项式的一般线性组合而出现的系统的基轨迹之外的解的数量，从而确定 Alt 问题的上限。特别是，我们推导出有效的符号和数值方法来研究使用概率饱和度的此类系统，这些方法可以使用有限域和浮点计算来使用。我们给出了实现所需确定性水平所需的有限域大小的界限。这些方法也可以应用于出现类似系统的许多其他问题，例如计算 Newton-Okounkov 体的体积和计算交叉理论不变量，包括 Euler 特征、Chern 类和 Segre 类。