This conflation is supported by the identity theorem for polynomials, that asserts that if two polynomial forms agree at an infinite number of (say) complex numbers, thus for infinitely many, then they agree as polynomial forms (i.e., their coefficients match). This definition (2) looks so similar to the definition (1) that we usually abuse notation and conflate with. The collection of such polynomial forms is denoted, and is a commutative ring.Ī polynomial form can be interpreted in any ring (even non-commutative ones) to create a polynomial function, defined by the formulaįor any. Where are coefficients in the integers, and is an indeterminate: a symbol that is often intended to be interpreted as an integer, real number, complex number, or element of some more general ring, but is for now a purely formal object. A polynomial form in, say, one variable with integer coefficients, is a formal expression of the form In algebra we see a distinction between a polynomial form (also known as a formal polynomial), and a polynomial function, although this distinction is often elided in more concrete applications. In the end it was a very simple observation (and would be extremely pedestrian to anyone who works in an algebraic field of mathematics), but as I found this helpful to a non-algebraist such as myself, and I thought I would share it here in case others similarly find it helpful. However, recently when pondering the very concrete concept of a polynomial, I managed to accidentally stumble upon a special case of Yoneda’s lemma in action, which clarified this lemma conceptually for me. There is a popular MathOverflow post devoted to this question, with many answers that were helpful to me, but I still felt vaguely dissatisfied. As someone who had a relatively light graduate education in algebra, the import of Yoneda’s lemma in category theory has always eluded me somewhat the statement and proof are simple enough, but definitely have the “ abstract nonsense” flavor that one often ascribes to this part of mathematics, and I struggled to connect it to the more grounded forms of intuition, such as those based on concrete examples, that I was more comfortable with.
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