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1. starting article on a degree of a polynomial, since its entry in the disambiguation page degree linked to a nonexistent page.

Anonymous

• CommentRowNumber2.
• CommentAuthorJ-B Vienney
• CommentTimeNov 23rd 2022

Added the definition of homogeneous polynomial as a $p$ such that the $n^{th}$ derivative of $p$ is $n.p$.

• CommentRowNumber3.
• CommentAuthorJ-B Vienney
• CommentTimeNov 23rd 2022

It was false, deleted.

• CommentRowNumber4.
• CommentAuthorJ-B Vienney
• CommentTimeNov 23rd 2022
• (edited Nov 23rd 2022)

Corrected: homogeneous polynomials in one variable $x$ of degree $n$ are the ones such that $\partial_x (p).x = n.p$.

• CommentRowNumber5.
• CommentAuthorJ-B Vienney
• CommentTimeNov 23rd 2022

@Anonymous: Do you have a reference where such a definition appears? I find it very interesting.

• CommentRowNumber6.
• CommentAuthorJ-B Vienney
• CommentTimeNov 23rd 2022

Added homogeneous polynomials in a finite number of indeterminate + a notion of polynomial homogeneous with respect to some variable.

2. Changed minimum to a slightly different version involving the maximum in the one indeterminate version because the definition using maximum generalizes better to the multiple indeterminant version, and to constructive mathematics.

Anonymous

3. swapped derivative out for shift operator in the single indeterminant case, so that it works for all rings, not just rings of characteristic zero.

Anonymous

4. 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

5. readded definition in terms of derivatives, would be useful to link it up to Euler’s identity

Anonymous

• CommentRowNumber11.
• CommentAuthorUrs
• CommentTimeNov 24th 2022

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 changed “isProp modality” to “isProp predicate”: the modality would be propositional truncation, which is different.
• CommentRowNumber12.
• CommentAuthorUrs
• CommentTimeNov 24th 2022

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

6. switched language over to set theory language

Anonymous

7. 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