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Definition of polynomial from google's search engine dictionary:

an expression of more than two algebraic terms, especially the sum of several terms that contain different powers of the same variable(s).

I searched on what poly means, and have got this from etymology dictionary:

word-forming element meaning "many, much, multi-, one or more,"....

And from wikipedia, monomial is

a polynomial which has only one term.

If google is right, how can monomial (having one term) be a polynomial (having greater than two terms - from the above definition)? Or is google wrong? Or is Etymology dictionary wrong?

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  • $\begingroup$ polynomial is a general definition , like "tree". Monomial is specific like "oak tree". an oak tree is still a tree, just like a monomial is still a polynomial $\endgroup$ – The Integrator Jun 12 '18 at 14:53
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    $\begingroup$ A simple def is : "a polynomial is an expression ..." A formal def is "Let $R$ he a ring and let $R[x]$ denote the set of all sequences of elements of $R$ : $(a_0,a_1,\ldots)$ such that $a_i = 0$ for all but a finite number of indices $i$." $\endgroup$ – Mauro ALLEGRANZA Jun 12 '18 at 14:54
  • $\begingroup$ I’ve always seen that $P_n(x) = \displaystyle \sum_{i=0}^n a_ix^i$ for $n\in\Bbb Z$ where $\bigcup_{i=0}^n\{a_i\}\subset\Bbb R$ or $\subset\Bbb C$, and if $n=0$ then so be it, though I can’t substantiate this. $\endgroup$ – gen-ℤ ready to perish Jun 12 '18 at 15:01
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The definition I learned for polynomials definitely included monomials. It also included constants and $0$. It would be very inconvenient to exclude them because the set of polynomials would not be closed under addition. The etymology is correct, but we don't have to follow etymology exactly when we define terms. I would see the poly- prefix as reflecting the fact that there may be many terms, but not requiring them.

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Don't rely on a dictionary for colloquial (or natural) language to be an accurate source of mathematical terminology. There are plenty of times that a mathematical term has a technical meaning which is not what you might expect based on the "usual" interpretation of the word or its parts. In the case of "polynomial" this is a simple instance of weakening, where "poly"is interpreted as "possibly many" rather than "many" (similarly to the use of "or" to mean inclusive "or" in logic); here are many more drastic examples of this, where the connection between term and meaning is tenuous or even nonexistent, my personal favorite being the use of the word "weasel" in set theory.

Wikipedia's definition of polynomial is in fact correct. (That's not to say that wikipedia is infallible, but it's pretty good generally for mathematics; one should really confirm wikipedia's definition by looking at an appropriate text.)


Incidentally, just to drive the point home: weasels are so called because they are like mice, but larger. Why are mice called that? To quote a famous philosopher, "I'll tell you ... I don't know."

(There are two competing stories that I've heard. One is that the mathematician in question, Jensen, made a typo and "we call such sets nice" turned into "we call such sets mice," and he liked it so much he kept it; the other is that Jensen simply wanted a word that had no prior meaning in mathematics and picked it mostly at random. My understanding is that Jensen himself has told both stories at different times.)

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