  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.
