Rational functions¶

Definition F-1. A rational function over \(\mathbb Q\) is an element of the fraction field \(\mathbb Q(X)\) of \(\mathbb Q[X]\). Its normalized form is the pair \((p, q)\) with \(p/q\) the element, \(q \neq 0\), \(\gcd(p, q) = 1\), and \(q\) with leading coefficient \(1\) in the chosen monomial order (and \(q = 1\) when \(p = 0\)).

Proposition F-1. Every rational function has exactly one normalized form.

Proof. If \(p/q = p'/q'\) in normalized forms, \(pq' = p'q\); since \(\gcd(p, q) = 1\), \(q \mid q'\), and symmetrically \(q' \mid q\), so they are associates; both being monic, \(q = q'\), and then \(p = p'\). Existence: divide any representation by the gcd of numerator and denominator and by the denominator's leading coefficient. \(\square\)

Theorem F-2 (cancellation). If \(p = g p'\), \(q = g q'\) (Certificate G-C1) and \(q \neq 0\), then \(p/q = p'/q'\) in \(\mathbb Q(X)\). Pointwise, for every point \(a\) with \(q(a) \neq 0\), also \(q'(a) \neq 0\) and \(p(a)/q(a) = p'(a)/q'(a)\).

Proof. In the field, \(p/q = (g p')/(g q') = p'/q'\) since \(g \neq 0\). At a point, \(q(a) = g(a) q'(a) \neq 0\) forces both factors nonzero. \(\square\)

The pointwise statement is what a CAS answer needs: in the scalar profile it is the transfer principle (Theorem X-6 of the kernel's theory pages) with the guard \(q \neq 0\) and the finiteness of the variables. Points where \(q' \ne 0\) but \(q = 0\) (the removed singularities, such as \(x = 1\) in \((x^2 - 1)/(x - 1)\)) are exactly where the original and the cancelled forms disagree as functions on the carrier, which is why polynorm states the guard \(q \neq 0\), not \(q' \neq 0\).

Proposition F-3 (arithmetic). For normalized \(p_1/q_1\) and \(p_2/q_2\), the normalized forms of their sum and product are obtained by forming \(p_1 q_2 + p_2 q_1\) over \(q_1 q_2\), respectively \(p_1 p_2\) over \(q_1 q_2\), and normalizing; the gcd can be computed more cheaply from \(\gcd(q_1, q_2)\) (Henrici's method) without changing the result, by Proposition F-1. \(\square\)

Certificate F-C1 (normalization). Data: the cofactors of G-C1 for the cancellation, and optionally G-C2 or G-C3 showing that the result is fully reduced. The claim needed for an equation is Theorem F-2's; full reduction is a quality claim.

The array facade requires G-C3 for every normalized function, so poly.rational.check(f) proves full reduction as well as the cancellation identity and monic denominator. The same argument applies over QQi, AA, and QQbar. The rational-function reference lists the multivariate coprimality and partial-fraction irreducibility checks.