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Let $X$ be the set of functions $e^{p(x)}$ of the real vector $x$, where $p$ is a multivariate polynomial with $p(0)=0$.

Is any finite subset of $X$ linearly independent? If yes, why? If no, is the answer true for other, restricted choices of $p$? (The answer is yes when the polynomials are restricted to have degree 1.)

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    $\begingroup$ What does applying a polynomial to a vector mean in this context? Does it just mean applying a multivariate polynomial to a given set of values? $\endgroup$ Feb 27, 2018 at 13:24
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    $\begingroup$ If these are univariate polynomials, then the set is indeed linearly independent, which is easily seen by considering the behaviour of a linear combination as $x\to\infty$ (one of the polynomials will dominate). $\endgroup$ Feb 27, 2018 at 14:31
  • $\begingroup$ @DanielMcLaury: yes; corrected $\endgroup$ Feb 27, 2018 at 15:01
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    $\begingroup$ Of course, as a minor matter of terminology, "$X$ is linearly independent" (for $X$ a subset of an abstract vector space) means "every finite subset of $X$ is linearly independent". $\endgroup$
    – LSpice
    Feb 27, 2018 at 19:18
  • $\begingroup$ For an elementary proof in the univariate case look here $\endgroup$ Oct 28, 2018 at 8:01

1 Answer 1

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Yes. If you require that no difference $p_j-p_k$ is constant (which follows from your assumption $p(0)=0$), then $c_1e^{p_1}+\ldots+c_me^{p_m}=0$ implies that all $c_j=0$.

In fact a more general result is true: instead of polynomials one can take any entire functions. This is called Borel's theorem. (See, for example, S. Lang, Introduction to complex hyperbplic spaces, Springer 1987, Theorem 1 in VII, sect 1, p. 186.)

For polynomials in one variable, this is easy: look at the asymptotics, for example when $z$ tends to infinity on an appropriate ray, and use induction in degree of the polynomials. To obtain the same for several variables, restrict your identity onto lines in $C^n$.

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  • $\begingroup$ I cannot find a Theorem 1 in Chapter VI, Section 1 (about the Poisson-Jensen formula in Nevanlinna theory). Please point to the page! $\endgroup$ Feb 28, 2018 at 10:46
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    $\begingroup$ @Arnold Neumaier: Sorry it is Chapter VII, page 186. $\endgroup$ Feb 28, 2018 at 13:41

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