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@Anonymous: Do you have a reference where such a definition appears? I find it very interesting.
also, moving query boxes to the nForum
+– {: .query} I might want to switch out derivatives with shift operators, since the definition involving the derivatives only work for commutative rings with characteristic zero. =–
+– {: .query} J-B: Do you know the notion of Hasse-Schmidt derivative? Maybe it could work in positive characteristic.
With one indeterminant $x$, it is defined for every $k \ge 0$ by $D^{k}(x^{n+k}) = \binom{n+k}{k} x^{n}$ and I think that we then have for all $P:R[x]$, $deg(P) = min \{i \ge 0, \forall l \ge i, D^{l}(P) = 0 \}$ (or something similar). =–
Anonymous
I have made adjustments to the wording in the beginning (here) of the Definition-section. Mostly just for readability, such as to reduce frequent repetitions, but also concerning one minor technical point:
I moved this claim from before to after the informal motivation (now here):
In the following, we work in dependent type theory with excluded middle:
But I am not sure what this comment is really doing: The material that follows does not look like it uses dependent type theory much at all. (?)
The only relevant comment I see is that rings of polynomials are advertized as “higher inductive types”. At that point it seems you really want to invoke a universal property
I’ve replaced all the instances of the nonce word ‘indeterminant’ with ‘indeterminate’ (Wiktionary and the sites that scrape it know the first word, but the second word has actual attestations and can be found in dictionaries.
Anonymous
I don’t really mean to rain on anyone’s parade, but to me the entry (along with some others it’s connected to) looks bloated or overworked or too wordy, or something like that, without much payoff.
I’m reading the fourth paragraph under the section heading “Definition”, where some reasons for what is about to follow are given, including a desire to avoid ellipses and indices and the like. I do sympathize with that desire. (For example, Eilenberg once said, “If you do things right, you won’t need subscripts”.) Nevertheless, I feel that something is a little off – too restrictive, perhaps – in how the issue is handled. I’d like to try to float some ideas different from the ones explored in the current article, and try especially to emphasize universal properties, in keeping with the nPOV.
A polynomial ring $R[x_1, \ldots, x_n]$ may be described as the free $R$-module generated by the set of monomials, i.e., the (underlying set of the) free commutative monoid $exp(n)$ on an $n$-element set. Notice that $exp(1) = \mathbb{N}$. Already on $exp(n)$ there is a “canonical” $\mathbb{N}$-valued degree function, $deg: exp(n) \to exp(1)$, namely the function obtained by applying the functor $exp$ to the unique function $n \to 1$. Of course there are many commutative monoid maps $w: \exp(n) \to \exp(1)$: by the universal property, these are in natural bijection with functions $n \to \exp(1)$ from an $n$-element set, where we think of such a function as assigning a weight to each of the $n$ elements. Note that the need to assign different weights to variables comes up frequently in practice, for example, when we consider various types of characteristic classes.
Next, let’s consider the free $R$-module construction on a set $S$, which I will denote as $R \cdot S$. Elements of $R \cdot S$ are given by finitely supported functions $S \to R$. (For the moment let me table the discussion of what “finitely supported” should mean in a constructive context.) Classically, the collection of finite subsets of $S$, which I will denote as $K(S)$, is the free commutative idempotent monoid on $S$. In this notation, the support function takes the form
$\supp: R \cdot S \to K(S).$Meanwhile, $\mathbb{N} = \exp(1)$ itself carries an obvious commutative idempotent monoid structure, namely the one whose “multiplication” is $(m, n) \mapsto \max\{m, n\}$. By the universal property, there is a canonical map of commutative idempotent monoids $\varepsilon: K(exp(1)) \to exp(1)$ that extends the identity function on $exp(1)$. Putting all this together, the degree function on polynomials with coefficients in $R$ attached to a weighting $w: exp(n) \to exp(1)$ is the composite
$R \cdot \exp(n) \stackrel{\supp}{\to} K(\exp(n)) \stackrel{K(w)}{\to} K(exp(1)) \stackrel{\varepsilon}{\to} \exp(1).$Of course, the $n$ here, standing for an $n$-element set of variables, can be replaced by an arbitrary set, and everything would go through without change. And look: not an ellipsis or subscript in sight in this description. Finally, there is a kind of built-in modularity: instead of polynomial algebras which are monoid algebras for the monoid $\exp(n)$ of monomials, we could consider monoid algebras for other sorts of monoids $M$ in place of $exp(n)$, and other sorts of weighting functions $w: M \to N$, where $N$ is a monoid bearing a secondary structure of commutative idempotent monoid (or join-semilattice structure). It seems to me lots of extensions of the degree concept are possible in this sort of framework.
(Although to be upfront, maybe I should worry that this apparently assigns to the zero element a degree of zero, which is not my preference! Hmm…)
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