Are there some books or articles regarding "polynomials" with non-integer (real) exponents, i.e., $$f(x)=a_1x^{e_1}+a_2x^{e_2}+\dots+a_nx^{e_n},$$ where $e_1,e_2,\dots$ are any real numbers (and $x$ being also real)?

I am mostly interested in theorems regarding the location of the roots, number of the roots, bounds, etc., of such "polynomials". Thanks.


2 Answers 2


I think the keyword is exponential polynomial because $x^\alpha=e^{\alpha \log x}$.

Try these papers:

  • 1
    $\begingroup$ This is a little misleading since the sort of thing pisoir is looking for would be an exponential polynomial in $\log x$, not in $x$. The third bullet point seems to provide "generalized polynomials", so maybe that's the most appropriate name? In this comment, kimchi-lover suggests the little-attested signomial as well. $\endgroup$
    – Mark S.
    Apr 10, 2023 at 1:38

One term for this is signomial, where the argument is restricted to the positive reals, to avoid definition problems of things like $x^{\sqrt 2}$ for negative $x$, and so on. I hope this term is not in widespread use, but the Wikipedia page does refer to a textbook that uses it.


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