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Given $\epsilon\in(0,1)$, suppose we have collection $\mathscr{C}(\epsilon)$ of multilinear polynomials in $\Bbb R[x_1,\dots,x_n]$ that on $\{0,1\}^n$ is in range $[-\epsilon,\epsilon]$ on $S_0$ while being in range $[1-\epsilon,1+\epsilon]$ on $S_1$ where $S_0\cap S_1=\emptyset$ with $S_0\cup S_1=\{0,1\}^n$.

Is this collection compact?

Turbo
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  • Which is the topology? – ajotatxe May 24 '15 at 09:18
  • Consider coefficients of polynomial in Euclidean space. – Turbo May 24 '15 at 09:27
  • Made an attempt on closedness. Define $g(p)=\max_{\underline{x}\in{0,1}^n}(p(\underline{x})-1_{S_1}(\underline{x}))$ where $1_{S_1}(\underline{x})=1$ if $\underline{x}\in S_1$ else $0$. Consider a sequence of polynomials ${p_i}_{i=0}^\infty$ in $\mathscr{C}(\epsilon)$ converging to $p$ in $\overline{\mathscr{C}(\epsilon)}$ (closure of collection $\mathscr{C}(\epsilon)$). – Turbo May 24 '15 at 09:28
  • We have sequence ${g(p_i)}_{i=0}^\infty$ converging to $g(p)\in[-\epsilon,\epsilon]\cup[1-\epsilon,1+\epsilon]$ since $[-\epsilon,\epsilon]\cup[1-\epsilon,1+\epsilon]$ is closed. Hence $p\in\mathscr{C}(\epsilon)$. So ${\mathscr{C}(\epsilon)}$ is closed. – Turbo May 24 '15 at 09:28

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