And this function is both additive and multiplicative (it’s a ring morphism from $R$ to $A$). That is, once you’ve chosen $a$, then $f(a)$ makes sense as a real number (as an element of $A$) no matter what polynomial $\,f$ you look at.įurthermore, the choice of $a$, once it’s done, gives you a function from polynomials to constants, I’ll call it $e_a$, namely $e_a(f)=f(a)$. Just to speak only of polynomials in one variable, the set of all such, $\mathbb R$ ($R$ for a general ring), has the property that the variable $x$ may be evaluated to any real number $a$ (to any element $\alpha$ of an algebra $A$ over the base ring $R$) so that this “evaluation mapping” can be applied to any polynomial at all (to any element of $R$ at all). The set (ring, actually) of polynomials with real coefficients (more generally with coefficients in any commutative ring) has a “universal” property that the larger sets do not. I’ll try to make my explanation as elementary as I can, with parenthetical expansions for the more technically inclined. Here’s my understanding of the reason for the restrictiveness of the definition of “polynomial”, in response to suggestion. I would opine that the reason that they're not the primary object of study is because they're not the 'simplest' structure of interest among any of their peers, and fundamentally the most important structures in mathematics tend to be the simplest structures exhibiting some given property.
As that article suggests, they're of particular importance and interest for their connections with the field of Hopf Algebras (and by extension, quantum groups). What's more, there's also an object occasionally studied that more directly corresponds to your notion: the notion of Laurent Polynomial.
Polynomials are defined as they are for a few distinct reasons: (1) because polynomials as functions have certain properties that your 'polynomials with division' don't have, and (2) because there are other terms for more generalized algebraic forms.įirst, the properties of polynomials: unlike e.g., $2x^$.
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