Lars V. Ahlfors (auth.), James D. Buckholtz, Teddy J.'s Complex Analysis: Proceedings of the Conference held at the PDF

By Lars V. Ahlfors (auth.), James D. Buckholtz, Teddy J. Suffridge (eds.)

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Extra info for Complex Analysis: Proceedings of the Conference held at the University of Kentucky, May 18–22, 1976

Example text

Though Ritt's results may not be extendable to rationals, nevertheless, analogues of I, III and IV in C do hold for rationals. 54 I. If P and Q are two polynomials without common zeros and each is either linear or of prime degree, then P/Q is a prime rational. One can show further by means of a parameter argument, a little less obvious than the one we used for polynomials, that for any composite number n there exist polynomials P and Q of degree n such that P/Q is prime. On the other hand one can readily verify that there exist nonlinear polynomials P1 and Qt and nonlinear polynomials P2 and Q2 such that with Ri(z)= Pi(z)/Qi(z) (i = 1,2), the rational function R 1 o R2 = P/Q, where P and Q are prime polynomials.

Let A(r'f) = fflz i~

If P(z) has two non-equivalent factorizations, one can pass from either to a factorization equivalent to the other by repeated steps of the three types indicated in VI. I through VII above pretty much sum up the essential facts about polynomial factorization. We now proceed to a discussion of the rational functions. D. Rational Functions. An analogous discussion for the factorization for rational functions is much more difficult. There is a much greater variety of possibilities. Ritt [30] claims that there are even cases in which the number of prime factors in one factorization is different from that in another.

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