is there a notion of algebraic (analytic) function?
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From: David Bernier on 26 Jun 2010 19:54 By analogy with algebraic numbers over the field Q of rational numbers, is there a notion of a complex-analytic function that is algebraic, that is some complex-analytic f, f: U -> C where U is a non-empty open subset of the field C of complex numbers which in some sense is "algebraic" ? (Informally, one speaks of transcendental functions, although I don't know of it as a mathematical concept). For example, polynomial functions in C[z] with coefficients being algebraic numbers would be algebraic functions. The quotient of two algebraic functions would be algebraic on any non-empty open sub-domain where the quotient is non-zero. The functions that are identically equal to an algebraic number would be algebraic. There would be closure under addition, multiplication operations over two algebraic functions, as long as their domain intersect on a non-empty open set. Applying the inverse function theorem would be allowed, to get a branch of z^(1/2) from f(z) = z^2. Composition of algebraic functions where the domain of the composition has as a subset a non-empty open set would be ok. Analytic continuation might be allowed. Integration couldn't always be allowed, or else taking a primitive of z -> 1/z would give z -> log(z) + C, and setting C = 0 would give z -> log(z). David Bernier
From: W. Dale Hall on 26 Jun 2010 21:08 David Bernier wrote: > By analogy with algebraic numbers over the field Q of rational numbers, > is there a notion of a complex-analytic function that > is algebraic, that is some complex-analytic f, > f: U -> C where U is a non-empty open subset of the field C of > complex numbers which in some sense is "algebraic" ? > > (Informally, one speaks of transcendental functions, > although I don't know of it as a mathematical concept). > > For example, polynomial functions in C[z] with coefficients being algebraic > numbers would be algebraic functions. The quotient of two algebraic > functions would be algebraic on any non-empty open sub-domain where > the quotient is non-zero. The functions that are identically equal > to an algebraic number would be algebraic. > > There would be closure under addition, multiplication operations over > two algebraic functions, as long as their domain intersect on > a non-empty open set. > > Applying the inverse function theorem would be allowed, > to get a branch of z^(1/2) from f(z) = z^2. > > Composition of algebraic functions where the domain of the > composition has as a subset a non-empty open set would be ok. > Analytic continuation might be allowed. > > Integration couldn't always be allowed, or else taking a > primitive of z -> 1/z would give z -> log(z) + C, > and setting C = 0 would give z -> log(z). > > David Bernier I'd have hoped this would have been obvious, but never mind my own prejudices. Take C, the field of complex numbers Form C[x] the ring of polnomials over C. It's a pretty well-behaved object, being a principal ideal domain and all Form C(x), the field of quotients of C[x]. These are called rational functions. Form Cbar(x), the algebraic closure of C(x). These will be algebraic functions. Since I made up the notation on the spot, I'd not be surprised if it failed to be the standard notation. Is that good enough for you? I suspect that the elements of Cbar(x) are all analytic, since you should be able to differentiate implicitly. I wouldn't be surprised either way with regard to the closure of Cbar(x) under the composition of functions. Just my two bits' worth, Dale
From: David Bernier on 27 Jun 2010 02:05 W. Dale Hall wrote: > David Bernier wrote: >> By analogy with algebraic numbers over the field Q of rational numbers, >> is there a notion of a complex-analytic function that >> is algebraic, that is some complex-analytic f, >> f: U -> C where U is a non-empty open subset of the field C of >> complex numbers which in some sense is "algebraic" ? >> >> (Informally, one speaks of transcendental functions, >> although I don't know of it as a mathematical concept). >> >> For example, polynomial functions in C[z] with coefficients being >> algebraic >> numbers would be algebraic functions. The quotient of two algebraic >> functions would be algebraic on any non-empty open sub-domain where >> the quotient is non-zero. The functions that are identically equal >> to an algebraic number would be algebraic. >> >> There would be closure under addition, multiplication operations over >> two algebraic functions, as long as their domain intersect on >> a non-empty open set. >> >> Applying the inverse function theorem would be allowed, >> to get a branch of z^(1/2) from f(z) = z^2. >> >> Composition of algebraic functions where the domain of the >> composition has as a subset a non-empty open set would be ok. >> Analytic continuation might be allowed. >> >> Integration couldn't always be allowed, or else taking a >> primitive of z -> 1/z would give z -> log(z) + C, >> and setting C = 0 would give z -> log(z). >> >> David Bernier > > I'd have hoped this would have been obvious, but never mind > my own prejudices. > > Take C, the field of complex numbers > > Form C[x] the ring of polnomials over C. It's a pretty > well-behaved object, being a principal ideal domain and all > > Form C(x), the field of quotients of C[x]. These are > called rational functions. > > Form Cbar(x), the algebraic closure of C(x). These will > be algebraic functions. Since I made up the notation on > the spot, I'd not be surprised if it failed to be the > standard notation. > > Is that good enough for you? All that is interesting. I can't picture the algebraic closure of C(x) too well. I found out about complex algebraic curves and a book on that subject by Frances Kirwan, "Complex algebraic curves", first published in 1992. Reference: < http://worldcat.org/identities/lccn-n84-77096 > (WorldCat Identities, Beta). David Bernier > I suspect that the elements of Cbar(x) are all analytic, since you > should be able to differentiate implicitly. I wouldn't be surprised > either way with regard to the closure of Cbar(x) under the > composition of functions. > > Just my two bits' worth, > Dale
From: G. A. Edgar on 27 Jun 2010 06:37 An function f analytic on a (connected) domain E is *algebraic* iff there is a nontrivial polynomial P of two variables such that P(z,f(z))=0 identically on E. And yes, log z or sin z are not algebraic on any domain. -- G. A. Edgar http://www.math.ohio-state.edu/~edgar/
From: Frederick Williams on 27 Jun 2010 09:20
David Bernier wrote: > > By analogy with algebraic numbers over the field Q of rational numbers, > is there a notion of a complex-analytic function that > is algebraic, that is some complex-analytic f, > f: U -> C where U is a non-empty open subset of the field C of > complex numbers which in some sense is "algebraic" ? Chapter 19 of van der Waerden's Algebra discusses algebraic function fields (over fields with a valuation generally). -- I can't go on, I'll go on. |