User:Chenxlee/TransHistory

The fundamental theorem of algebra implies that a polynomial of degree d with real coefficients has at most d real roots. This upper bound can be very tight, for example the quadratic polynomial x² − 3x + 2 does indeed have two real roots, at x = 1 and x = 2. In some situations this upper bound is much too high, though. The polynomial x⁹⁹ − 1, for example, only has one real root, specifically x = 1. The other ninety eight roots are non-real complex numbers.

In situations like this last one, a better bound for the number of positive real roots can be determined using Descartes' rule of signs. This rule says that the degree of the polynomial is irrelevant when counting the number of real roots, instead it is the number of sign changes between successive non-zero monomials that matters. So the polynomial x⁹⁹ − 1 only has one sign change, hence has one positive real root. This method only bounds the number of positive real roots, but making the substitution y = −x and repeating the procedure also bounds the number of negative roots of the original polynomial. In particular, then, the number of real roots in total is bounded above by 2r − 1, where r is the number of monomials that appear, since there can be at most r − 1 sign changes in the original polynomial, and at most r − 1 sign changes once the substitution y = −x is made, and finally there may be the solution x = 0.

• A. G. Khovanskii, On a class of systems of transcendental equations, Soviet Math. Dokl. 22 (1980), pp.762–765.