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I have read that the constant term of a univariate polynomial is assumed to be the constant coefficient multiplied with the variable raised to zero (which makes the variable equal one for any input other than zero and effectively makes the entire term equal the constant that is the coefficient.) It makes sense for all inputs other than zero, but zero raised to zero is not defined so doesn't that cause problems about how we define polynomials? If it does, what exactly are they (I cannot put my finger on it altough I suspect there has to be some problems) and how are they avoided?

jacob78
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    We have problems in context where quantities tend to zero. When we raise zero to exactly zero, there aren't any problems of definition. I suggest you to take a read – Turquoise Tilt Dec 08 '23 at 21:20
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    In the interest of writing polynomials with a less cumbersome notation, we simply assume conventionally $0^0=1$ when writing a polynomial in the form $\sum_{k=0}^n a_k x^k$. – PrincessEev Dec 08 '23 at 21:20
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    Where have you read this? It is incompatible with the usual reading of $t^0$ for specific numeric values of $t$ (namely $1$ when $t = 0$ and $0$ otherwise). In an abstract ring of polynomials over an indeterminate $x$, $x^0$ is not reallly meaningful, but it is convenient to think of the term of degree zero to be the constant term and hence formally treat $x^0$ as $1$. – Rob Arthan Dec 08 '23 at 21:26
  • @RobArthan apparently it is one of those cases where certain sources distort mathematical concepts for the sake of "simplicity". I was unaware of this convention and now that it is resolved for me, should I delete the question since it seems unnecessary? – jacob78 Dec 08 '23 at 21:39
  • I would leave the question up. It and the comments may help others. – Rob Arthan Dec 08 '23 at 21:43
  • "(namely 1 when t=0 and 0 otherwise)" , isn't there a typo anywhere ? For $t\ne 0$ , we have $t^0=1$ , not $t^0=0$ – Peter Dec 09 '23 at 07:16

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