Degree of the zero polynomial

The zero polynomial has no nonzero term, so its degree is either left undefined or assigned a special value such as $-\infty$ by convention. A finite program can use $-1$ as a sentinel when all other stored degrees are nonnegative. That representation sentinel is not a general replacement for the extended-arithmetic degree convention; it is enough for testing whether a polynomial remainder is zero or has smaller degree than a nonzero divisor.