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The author of the book I am referring assumes two rational integral functions as shown below

$p_0\cdot x^n+p_1\cdot x^{(n-1)}+p_2\cdot x^{(n-2)}+...+p_n$

$q_0\cdot x^n+q_1\cdot x^{(n-1)}+q_2\cdot x^{(n-2)}+...+q_n$

Is this term old and he means real functions ?; if not, what is meant by rational integral function ?, rational function, integral function ?

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    $\begingroup$ Probably just means "polynomial". Or maybe, "polynomial with rational coefficients". $\endgroup$ – Gerry Myerson May 10 '15 at 6:22
  • $\begingroup$ @GerryMyerson, thankx $\endgroup$ – Vikram May 10 '15 at 6:25
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    $\begingroup$ This book also defines the term to mean 'polynomial' books.google.ca/… $\endgroup$ – user142523 Nov 2 '17 at 22:15
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The heading of chap.4 of Modern Algebra 1 (1949) by Van der Waerden is "Polynomials" in English, and "Ganze rationale Funktionen" in German (http://link.springer.com/chapter/10.1007%2F978-3-662-41906-9_5). According to Bieberbach (Algebra, 1928) a rational analytic expression is an integral rational function if there are no variable divisors.

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I think this would be helpful:

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Reference: Handbook of Mathematics and Computational Science By John W. Harris, Horst Stöcker.

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