Within a few years of 1980, problems from three different engineering departments at GM Research—Computer Science, Engine Research, and Power Systems—which had come up in three entirely different applications were brought to the Mathematics Department to see if we could help solve them. Each problem reduced to finding all solutions to a system of polynomial equations.

Although the engineers were aware of many methods for solving many kinds of systems of equations, there was at that time no standard practical method for finding all solutions to systems of polynomial equations. The favored approach for several of these engineering groups—reducing the system to a single polynomial—can be particularly challenging to implement, even if it works “in theory.” A specific example will be given that illustrates this point.

The polynomial continuation method we eventually used was available in the recent mathematics literature. (The earliest paper I can find is from 1977.) But the engineers in these departments would never have found the method they needed in this literature, because: the outlets used by the mathematicians (books and journals) were unlikely to be looked at by the engineers, who would tend to look in “their own” literature. For the same reason, successful mathematical methods for one area of engineering might not be discovered by engineers in a different area; the language the engineers used to describe their problems was radically different from the language used in the mathematics literature: polynomial systems, homotopies, continuation, ideals, functional differential equations, fixed points, degree theory, etc.; The mathematics used to justify the methods—concerning either piecewise linear or smooth manifolds in Euclidean space—was unfamiliar to these engineers; the ultimate effectiveness of a method described in a paper is difficult to judge for a particular application. “Small” issues can make a method very challenging to apply or worthless.

The story here, then, is how these barriers were overcome at General Motors leading to three successful applications in three different departments and also helping to inspire the broadly useful and recognized area of applied algebraic geometry.