rational function (changes) in nLab

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Definition

As partial functions on a field

If kk is a field, given the polynomial ring k[x]k[x], there is a canonical ring homomorphism j:k[x]→(k→k)j:k[x] \to (k \to k) which sends constant polynomials in k[x]k[x] to constant functions in k→kk \to k, and the generator x∈k[x]x \in k[x] to the identity function id kid_k, where k→kk \to k is the function algebra on kk.

There is a function

i:k(x)→{A∈Ob(Part(k))|Hom(A,k)}i:k(x) \to \{A \in Ob(Part(k)) \vert Hom(A, k)\}

from the field of rational expression to the set of all partial functions in the category of partial functions Part(k)Part(k), where the function algebra k→kk \to k is the endomorphism kk-algebra on the improper subset kk. The function ii is defined as follows: given two polynomials p,q∈k[x]p, q \in k[x] , for where allx∈{a∈k|q(a)≠0} x q \in \{a \in k | q(a) \neq 0\} 0, for all x∈{a∈k|q(a)≠0}x \in \{a \in k | q(a) \neq 0\}

i(pq)(x)≔j(p)(x)j(q)(x)i\left(\frac{p}{q}\right)(x) \coloneqq \frac{j(p)(x)}{j(q)(x)}

where forx∈kx \in k and y∈ky \in k

xy\frac{x}{y}

is the division of xx by yy in kk . , A and forrational functionr∈k(x)r \in k(x) and f s∈k(x) f s \in k(x) , is anelement of the image of ii, f∈im(i)f \in \im(i).

rs\frac{r}{s}

is the division of rr by ss in k(x)k(x). A rational function ff is an element of the image of ii, f∈im(i)f \in \im(i).

As functions on the projective line

It is perhaps more illuminating to think of this partial function (with domain DD) as coming from a (total) function [p/q]:ℙ 1(k)→ℙ 1(k)[p/q]: \mathbb{P}^1(k) \to \mathbb{P}^1(k) on the projective line, where we have inclusion functions i:k→ℙ 1(k)i: k \to \mathbb{P}^1(k) and the partial function is given by the pair of projection maps in the pullback

D → k ↓ ↓ i k →[p/q]∘i ℙ 1(k)\array{ D & \to & k \\ \downarrow & & \downarrow_\mathrlap{i} \\ k & \underset{[p/q] \circ i}{\to} & \mathbb{P}^1(k) }

It should also be noticed that such endofunctions [p/q][p/q] on ℙ 1(k)\mathbb{P}^1(k) are closed under composition (except that special provision must be made for the constant function valued at ∞\infty, which corresponds to the “fraction” 1/01/0).

Properties

Rational functions on a field do not form a field

This comes from the fact that the reciprocal function is not a multiplicative inverse of the identity function f(x)=xx≠1f(x) = \frac{x}{x} \neq 1, due to the fact that f(x)f(x) is undefined at 00, and thus has a different domain than the constant function 11, which is the multiplicative unit.

As continuous functions

rational functions are continuous on their domain of definition

• polynomial

• power series

• rational map

• homotopy of rational maps

• category of partial functions

• rational fraction

• real rational function?

• rational function on an affine variety

References

• Wikipedia, Rational function

On the homotopy type of spaces of rational maps from the Riemann sphere to itself (related to the moduli space of monopoles in ℝ 3\mathbb{R}^3 and to the configuration space of points in ℝ 2\mathbb{R}^2):

• Graeme Segal, The topology of spaces of rational functions, Acta Math. Volume 143 (1979), 39-72 (euclid:1485890033)