# Probabilistic Saturations and Alt's Problem

## TL;DR

This paper develops probabilistic symbolic and numeric methods using saturations to estimate upper bounds for solutions to Alt's problem, a classical problem in kinematics, with applications to algebraic geometry.

## Contribution

It introduces effective probabilistic saturation techniques for counting solutions to polynomial systems, providing bounds and methods applicable beyond Alt's problem.

## Key findings

- Derived bounds on the number of solutions outside the base locus.
- Developed methods using finite fields and floating-point computations.
- Applicable to problems in algebraic geometry like Newton-Okounkov bodies and intersection invariants.

## 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.