Consider the set $S$ of all infinitely-differentiable functions on the reals. Consider the structure $(S,+,-,*,0,1,Id,D)$, where $+$,$-$, and $*$ are function addition, subtraction and multiplication of functions, $0$ and $1$ are constant functions, $Id$ is the identity function on the reals, and D is the differentiation operator. Basically, my question is, is the sum, difference, and product rules, along with the rules that the derivative of $0$ and $1$ are $0$ and the derivative of $Id$ is $1$ sufficient to prove all universal equations true in that structure? And if not, is there a larger finite, or at worst recursive set of axioms that are sufficient to derive all universal equations? In other words, what I am really asking is if there are any further differentiation rules that have yet to be discovered.
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Besides the axioms involving $D$ that you mention, you probably also ought to include axioms that say that $(S,+,-,*,0,1)$ forms a commutative ring. – Eric Wofsey Jul 03 '15 at 08:19
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I guess limits are indispensable. – Eclipse Sun Jul 03 '15 at 08:19
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I believe the answer is yes, because $S$ contains $\mathbb{Z} [x]$, and it seems to me that $\mathbb{Z} [x]$ is the free commutative ring equipped with a derivation $D$ and an element $x$ such that $D x = 1$. – Zhen Lin Jul 03 '15 at 08:31
1 Answers
Let $T(f_1,\dots,f_n)$ be any term in $n$ variables in this language. By an easy induction, your axioms (plus the axioms for a commutative ring) imply that $T$ is equal to some polynomial in $Id$ and the (iterated) derivatives of the $f_i$. To show that there are no other universal equations that hold in $S$, it thus suffices to show the following: if $P$ is a nonzero polynomial in $1+kn$ variables, then there exist $f_1,\dots f_n\in S$ such that $$P(Id, f_1,\dots,f_n, Df_1,\dots, Df_n,\dots, D^{k-1}f_1,\dots, D^{k-1}f_n)\neq 0.$$
But this is easy: there exist real numbers $r_0,\dots, r_{kn}$ such that $P(r_0,\dots,r_{kn})\neq 0$, and there exist $f_1,\dots, f_n\in S$ such that $D^jf_i(r_0)=r_{jn+i}$ for all $1\leq i\leq n$, $0\leq j<k$ (in fact, there exist such $f_i$ that are all polynomials of degree $<k$).
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