On Monday 04 December 2006 11:16, William Stein wrote: > On Mon, 04 Dec 2006 04:52:02 -0800, Joel B. Mohler > > <[EMAIL PROTECTED]> wrote: > > On Sunday 03 December 2006 10:23, William Stein wrote: > >> > If the predefined indeterminate belongs to a ring, which ring is it? > >> > >> Very likely it would just be ZZ[a,b,c,d...,z, A,B,C,...,Z]. > >> Alternatively, > >> they could be "formal indeterminates" that don't belong to some "formal > >> > >> ring", > >> but that's less easy to think about. > > > > What if you want to differentiate f(x)=x^x. This is senseless in a > > polynomial > > ring. > > This is the same as saying that "1/2" is senseless if 1 and 2 > are integers. > > This is not a problem. That the expression x^x does not define an element > of the polynomial ring, is not a problem. > Exponentiation of an element of the polynomial ring to the power > of a non-integer would be a constructor for some other sort of object, > namely a function in some general function space.
I'm not sure I entirely agree with this. I see that it could be done this way, but it seems quite mathematically strange to me. The fraction field of a ring is a well-known mathematical object, but the idea of putting a generator for a polynomial ring in an exponent is entirely undefined in any abstract sense (unless there's some mathematics I'm missing). -- Joel --~--~---------~--~----~------------~-------~--~----~ To post to this group, send email to [email protected] To unsubscribe from this group, send email to [EMAIL PROTECTED] For more options, visit this group at http://groups.google.com/group/sage-devel URLs: http://sage.scipy.org/sage/ and http://modular.math.washington.edu/sage/ -~----------~----~----~----~------~----~------~--~---
