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rational function

Contents

Definition

Given an integral domain RR, the commutative ring of rational functions R(z)R(z) with coefficients in RR is the field of fractions of the polynomial ring R[z]R[z].

Let XX be an affine variety over a field kk with the ring of regular function?s 𝒪(X)\mathcal{O}(X). A rational function is any element of the field of fractions of 𝒪(X)\mathcal{O}(X), that is the function field of the variety.

As “functions”

In either case, rational functions are equivalence classes of fractions; they need not be functions defined everywhere. If kk is a field, then to each fraction p(x)q(x)k(x)\frac{p(x)}{q(x)} \in k(x) with p,qk[x]p, q \in k[x] relatively prime, we may associate a partial function kkk \rightharpoonup k whose domain consists of aka \in k such that q(a)0q(a) \neq 0 (defining q(a)q(a) as usual as the value of qq under the unique kk-algebra map k[x]kk[x] \to k sending xx to aa), and sending each aa in the domain to p(a)q(a)\frac{p(a)}{q(a)}. For the purposes of most elementary mathematics, the domain given here may be described as the “natural domain” of the rational function.

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

As continuous functions

rational functions are continuous on their domain of definition

Function field analogy

function field analogy

number fields (“function fields of curves over F1”)function fields of curves over finite fields 𝔽 q\mathbb{F}_q (arithmetic curves)Riemann surfaces/complex curves
affine and projective line
\mathbb{Z} (integers)𝔽 q[z]\mathbb{F}_q[z] (polynomials, function algebra on affine line 𝔸 𝔽 q 1\mathbb{A}^1_{\mathbb{F}_q})𝒪 \mathcal{O}_{\mathbb{C}} (holomorphic functions on complex plane)
\mathbb{Q} (rational numbers)𝔽 q(z)\mathbb{F}_q(z) (rational functions)meromorphic functions on complex plane
pp (prime number/non-archimedean place)x𝔽 px \in \mathbb{F}_pxx \in \mathbb{C}
\infty (place at infinity)\infty
Spec()Spec(\mathbb{Z}) (Spec(Z))𝔸 𝔽 q 1\mathbb{A}^1_{\mathbb{F}_q} (affine line)complex plane
Spec()place Spec(\mathbb{Z}) \cup place_{\infty} 𝔽 q\mathbb{P}_{\mathbb{F}_q} (projective line)Riemann sphere
p() p()p\partial_p \coloneqq \frac{(-)^p - (-)}{p} (Fermat quotient)z\frac{\partial}{\partial z} (coordinate derivation)
genus of the rational numbers = 0genus of the Riemann sphere = 0
formal neighbourhoods
p\mathbb{Z}_p (p-adic integers)𝔽 q[[tx]]\mathbb{F}_q[ [ t -x ] ] (power series around xx)[[zx]]\mathbb{C}[ [z-x] ] (holomorphic functions on formal disk around xx)
Spf( p)×Spec()XSpf(\mathbb{Z}_p)\underset{Spec(\mathbb{Z})}{\times} X (“pp-arithmetic jet space” of XX at pp)formal disks in XX
p\mathbb{Q}_p (p-adic numbers)𝔽 q((zx))\mathbb{F}_q((z-x)) (Laurent series around xx)((zx))\mathbb{C}((z-x)) (holomorphic functions on punctured formal disk around xx)
𝔸 = pplace p\mathbb{A}_{\mathbb{Q}} = \underset{p\; place}{\prod^\prime}\mathbb{Q}_p (ring of adeles)𝔸 𝔽 q((t))\mathbb{A}_{\mathbb{F}_q((t))} ( adeles of function field ) x((zx))\underset{x \in \mathbb{C}}{\prod^\prime} \mathbb{C}((z-x)) (restricted product of holomorphic functions on all punctured formal disks, finitely of which do not extend to the unpunctured disks)
𝕀 =GL 1(𝔸 )\mathbb{I}_{\mathbb{Q}} = GL_1(\mathbb{A}_{\mathbb{Q}}) (group of ideles)𝕀 𝔽 q((t))\mathbb{I}_{\mathbb{F}_q((t))} ( ideles of function field ) xGL 1(((zx)))\underset{x \in \mathbb{C}}{\prod^\prime} GL_1(\mathbb{C}((z-x)))
theta functions
Jacobi theta function
zeta functions
Riemann zeta functionGoss zeta function
branched covering curves
KK a number field (K\mathbb{Q} \hookrightarrow K a possibly ramified finite dimensional field extension)KK a function field of an algebraic curve Σ\Sigma over 𝔽 p\mathbb{F}_pK ΣK_\Sigma (sheaf of rational functions on complex curve Σ\Sigma)
𝒪 K\mathcal{O}_K (ring of integers)𝒪 Σ\mathcal{O}_{\Sigma} (structure sheaf)
Spec an(𝒪 K)Spec()Spec_{an}(\mathcal{O}_K) \to Spec(\mathbb{Z}) (spectrum with archimedean places)Σ\Sigma (arithmetic curve)ΣP 1\Sigma \to \mathbb{C}P^1 (complex curve being branched cover of Riemann sphere)
() pΦ()p\frac{(-)^p - \Phi(-)}{p} (lift of Frobenius morphism/Lambda-ring structure)z\frac{\partial}{\partial z}
genus of a number fieldgenus of an algebraic curvegenus of a surface
formal neighbourhoods
vv prime ideal in ring of integers 𝒪 K\mathcal{O}_KxΣx \in \SigmaxΣx \in \Sigma
K vK_v (formal completion at vv)((z x))\mathbb{C}((z_x)) (function algebra on punctured formal disk around xx)
𝒪 K v\mathcal{O}_{K_v} (ring of integers of formal completion)[[z x]]\mathbb{C}[ [ z_x ] ] (function algebra on formal disk around xx)
𝔸 K\mathbb{A}_K (ring of adeles) xΣ ((z x))\prod^\prime_{x\in \Sigma} \mathbb{C}((z_x)) (restricted product of function rings on all punctured formal disks around all points in Σ\Sigma)
𝒪\mathcal{O} xΣ[[z x]]\prod_{x\in \Sigma} \mathbb{C}[ [z_x] ] (function ring on all formal disks around all points in Σ\Sigma)
𝕀 K=GL 1(𝔸 K)\mathbb{I}_K = GL_1(\mathbb{A}_K) (group of ideles) xΣ GL 1(((z x)))\prod^\prime_{x\in \Sigma} GL_1(\mathbb{C}((z_x)))
Galois theory
Galois groupπ 1(Σ)\pi_1(\Sigma) fundamental group
Galois representationflat connection (“local system”) on Σ\Sigma
class field theory
class field theorygeometric class field theory
Hilbert reciprocity lawArtin reciprocity lawWeil reciprocity law
GL 1(K)\GL 1(𝔸 K)GL_1(K)\backslash GL_1(\mathbb{A}_K) (idele class group)
GL 1(K)\GL 1(𝔸 K)/GL 1(𝒪)GL_1(K)\backslash GL_1(\mathbb{A}_K)/GL_1(\mathcal{O})Bun GL 1(Σ)Bun_{GL_1}(\Sigma) (moduli stack of line bundles, by Weil uniformization theorem)
non-abelian class field theory and automorphy
number field Langlands correspondencefunction field Langlands correspondencegeometric Langlands correspondence
GL n(K)\GL n(𝔸 K)//GL n(𝒪)GL_n(K) \backslash GL_n(\mathbb{A}_K)//GL_n(\mathcal{O}) (constant sheaves on this stack form unramified automorphic representations)Bun GL n()(Σ)Bun_{GL_n(\mathbb{C})}(\Sigma) (moduli stack of bundles on the curve Σ\Sigma, by Weil uniformization theorem)
Tamagawa-Weil for number fieldsTamagawa-Weil for function fields
theta functions
Hecke theta functionfunctional determinant line bundle of Dirac operator/chiral Laplace operator on Σ\Sigma
zeta functions
Dedekind zeta functionWeil zeta functionzeta function of a Riemann surface/of the Laplace operator on Σ\Sigma
higher dimensional spaces
zeta functionsHasse-Weil zeta function

Revised on May 8, 2017 02:52:05 by Urs Schreiber (92.218.150.85)