As with the inclusion of points at infinity and complexification of real polynomials, this allows some theorems to be stated more simply without exceptions and for a more regular algebraic analysis of the geometry.
12.
A "'proper rational function "'is a rational funcion in which the degree of P ( x ) is no greater than the degree of Q ( x ) and both are real polynomials.
13.
The non-real roots of a real polynomial with real coefficients can be grouped into pairs of complex conjugates, namely with the two members of each pair having imaginary parts that differ only in sign and the same real part.
14.
There are various proofs of this theorem, either by analytic methods such as topological ones such as the winding number, or a proof combining Galois theory and the fact that any real polynomial of " odd " degree has at least one real root.
15.
The separable permutations also have a characterization from algebraic geometry : if a collection of distinct real polynomials all have equal values at some number, then the permutation that describes how the numerical ordering of the polynomials changes at is separable, and every separable permutation can be realized in this way.
16.
Berg's result states that every non-negative real polynomial within a bounded interval can be approximated within accuracy \ epsilon on that interval with a sum-of-squares of real polynomials of sufficiently high degree, and thus if OBJ ( x ) is the polynomial objective value as a function of the point x, if the inequality c + \ epsilon-OBJ ( x ) \ ge 0 holds for all x in the region of interest, then there must be a sum-of-squares proof of this fact.
17.
Berg's result states that every non-negative real polynomial within a bounded interval can be approximated within accuracy \ epsilon on that interval with a sum-of-squares of real polynomials of sufficiently high degree, and thus if OBJ ( x ) is the polynomial objective value as a function of the point x, if the inequality c + \ epsilon-OBJ ( x ) \ ge 0 holds for all x in the region of interest, then there must be a sum-of-squares proof of this fact.
