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Assume $p$ and $q$ are n-variate degree d, homogeneous polynomials. Define $ |p|_{\infty}= \max_{x \in S^{n-1}} |p(x)|$ $D(p)=\{ x \in S^{n-1} : p(x)= |p|_{\infty} \}$
$E(p)=\{x \in S^{n-1} : p(x)= -|p|_{\infty} \}$

Define $D(q)$ and $E(q)$ similarly. Now assume $ D(p) \cup E(p) = D(q) \cup E(q) $ What kind of relation we can deduce between $p$ and $q$ by this assumption?

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  • $\begingroup$ A polynomial of even order can be bounded from one direction, but no nonconstant polynomial in the whole space is bounded. It would make a reasonable question to ask what is the relation of two such polynomials that reach their extremum in the same set. Would something like this work for you? $\endgroup$
    – Joonas Ilmavirta
    Commented Mar 17, 2015 at 6:50
  • $\begingroup$ I've changed the question to make more sense :) This version would be helpful for me. $\endgroup$
    – alpx
    Commented Mar 18, 2015 at 0:00
  • $\begingroup$ Do you happen to know of any examples where $p$ and $q$ are not in a linear relation? $\endgroup$
    – Yaakov Baruch
    Commented Mar 18, 2015 at 8:45
  • $\begingroup$ An answer to my question: if any $r$ has degree $e$ then $r^3$ and $r\big(\sum{x_i^2}\big)^e$ would have the same extreme sets, without being linearly related (in either $\mathbb{R}^n$ or $S^{n-1}$). $\endgroup$
    – Yaakov Baruch
    Commented Mar 18, 2015 at 12:37

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I think that some computing around shows that $(x^6+y^6)^5$ and $(x^{10}+y^{10})^3$, or $x^{12}+y^{12}$ and $(x^4+y^4)^3$ for that matter, have the same set of extremes. So there can't be an algebraic relation between $p$ and $q$ holding in general.

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  • $\begingroup$ @ Yaakov: Thanks for the answer. Could you please explain a little about your computations? Otherwise one could still write a generic algebraic relation with this polynomials. $\endgroup$
    – alpx
    Commented Mar 19, 2015 at 15:40
  • $\begingroup$ @alpx $(x^{2m}+y^{2m})/(x^2+y^2)^m$ has absolute max points at $(\pm 1, 0), (0, \pm 1)$ and absolute min points at $(\pm \frac{\sqrt{2}}{2}, \pm \frac{\sqrt{2}}{2})$. These are then max/min points on $S^1$ too since $x^2+y^2$ is 1 there. Raising to odd powers then only changes the values, not the locations, of extreme points. I think similar result hold in higher dimensions too. $\endgroup$
    – Yaakov Baruch
    Commented Mar 19, 2015 at 16:41

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