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nLab > Latest Changes: degree of a polynomial

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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

    v1, current

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

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

    diff, v2, current

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

    It was false, deleted.

    diff, v2, current

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

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

    diff, v2, current

    • 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.

    diff, v4, current

  7. 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

    diff, v8, current

  8. 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

    diff, v10, current

  9. 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 xx, it is defined for every k≥0k \ge 0 by D k(x n+k)=(n+kk)x nD^{k}(x^{n+k}) = \binom{n+k}{k} x^{n} and I think that we then have for all P:R[x]P:R[x], deg(P)=min{i≥0,∀l≥i,D l(P)=0}deg(P) = min \{i \ge 0, \forall l \ge i, D^{l}(P) = 0 \} (or something similar). =–

    Anonymous

    diff, v10, current

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

    Anonymous

    diff, v10, current

    • 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.

    diff, v11, current

    • 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

    diff, v11, current

  13. switched language over to set theory language

    Anonymous

    diff, v12, current

  14. 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

    diff, v12, current

  15. adding context sidebar

    Anonymous

    diff, v12, current

    • CommentRowNumber16.
    • CommentAuthorTodd_Trimble
    • CommentTimeJul 28th 2023

    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,…,x n]R[x_1, \ldots, x_n] may be described as the free RR-module generated by the set of monomials, i.e., the (underlying set of the) free commutative monoid exp(n)exp(n) on an nn-element set. Notice that exp(1)=ℕexp(1) = \mathbb{N}. Already on exp(n)exp(n) there is a “canonical” ℕ\mathbb{N}-valued degree function, deg:exp(n)→exp(1)deg: exp(n) \to exp(1), namely the function obtained by applying the functor expexp to the unique function n→1n \to 1. Of course there are many commutative monoid maps w:exp(n)→exp(1)w: \exp(n) \to \exp(1): by the universal property, these are in natural bijection with functions n→exp(1)n \to \exp(1) from an nn-element set, where we think of such a function as assigning a weight to each of the nn 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 RR-module construction on a set SS, which I will denote as R⋅SR \cdot S. Elements of R⋅SR \cdot S are given by finitely supported functions S→RS \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 SS, which I will denote as K(S)K(S), is the free commutative idempotent monoid on SS. In this notation, the support function takes the form

    supp:R⋅S→K(S).\supp: R \cdot S \to K(S).

    Meanwhile, ℕ=exp(1)\mathbb{N} = \exp(1) itself carries an obvious commutative idempotent monoid structure, namely the one whose “multiplication” is (m,n)↦max{m,n}(m, n) \mapsto \max\{m, n\}. By the universal property, there is a canonical map of commutative idempotent monoids ε:K(exp(1))→exp(1)\varepsilon: K(exp(1)) \to exp(1) that extends the identity function on exp(1)exp(1). Putting all this together, the degree function on polynomials with coefficients in RR attached to a weighting w:exp(n)→exp(1)w: exp(n) \to exp(1) is the composite

    R⋅exp(n)→suppK(exp(n))→K(w)K(exp(1))→εexp(1).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 nn here, standing for an nn-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)\exp(n) of monomials, we could consider monoid algebras for other sorts of monoids MM in place of exp(n)exp(n), and other sorts of weighting functions w:M→Nw: M \to N, where NN 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.

    • CommentRowNumber17.
    • CommentAuthorTodd_Trimble
    • CommentTimeJul 28th 2023

    (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…)

  18. changed higher algebra - contents to algebra - contents in context sidebar

    Anonymouse

    diff, v16, current

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