Polynomials in many variables

Some of the most important applications come from polynomials in several variables. Applications include robotics, error-correcting codes, cryptography, and geometry. We will only scratch the surface of this vast field in this book. Think back to the long division algorithm for polynomials in one variable. When dividing $f(x)$ by $g(x)$, one of the first things to do is to identify the leading, or highest order, term of $f(x)$ and the leading term of $g(x)$. The expression ``leading term of'' has not yet been defined for polynomials of several variables. Motivated by the hope to generalize the division algorithm from one to several variables, we next discuss how to define the ``leading term''.