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Real-valued functions of real variable, analytic properties of functions and sequences, limits, continuity, smoothness of these.

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Nowhere negative polynomials form a semialgebraic set

Let $P_{d, n}$ be the space of polynomial maps $\mathbb{R}^n\to \mathbb{R}$ of degree at most $d$. Is the subset $S\subset P_{d, n}$ of nowhere negative polynomials semialgebraic?