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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 R^n : p(x)= |p|_{\infty} \}$$D(p)=\{ x \in S^{n-1} : p(x)= |p|_{\infty} \}$
$E(p)=\{x \in R^n : p(x)= -|p|_{\infty} \}$$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?

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

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?

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?

I've changed question to make more sense :)
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Assume $p$ and $q$ are n-variate degree d, homogeneous polynomials. Define $ |p|_{\infty}= \max_{x \in R^n}|p(x)|$$ |p|_{\infty}= \max_{x \in S^{n-1}} |p(x)|$ $D(p)=\{ x \in R^n : p(x)= |p|_{\infty} \}$
$E(p)=\{x \in R^n : p(x)= -|p|_{\infty} \}$

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?

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

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?

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

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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polynomials with similar maxima-minima

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

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?