Consider a real quintic polynomial $$p(x;\, \alpha, \beta)=a_0 (\alpha,\beta) + a_1 (\alpha,\beta) x + a_2 (\alpha,\beta) x^2+ a_3 (\alpha,\beta) x^3 + a_4 (\alpha,\beta) x^4 - x^5$$ with real valued functions $a_i$ defined by $$\forall i \in \{1,\ldots, 5\}\quad a_i:\Omega \to \mathbb{R}, $$ where $\Omega \subset \mathbb{R}^2$.

I'd like to proof, that $p$ has only real roots in $x$ for all $(\alpha,\beta) \in \Omega$. A proof relying on Sturm's Theorem seems not feasible as the given functions $\alpha_i$ are quite complex expressions themselves. Is there an easier method to accomplish this?

  • $\begingroup$ How complex are the $a_i$? Are they at least continuous? $\endgroup$ – Hagen von Eitzen Feb 11 '13 at 13:14

I assume all $a_i$ are continuous. Compute the discriminant $D(\alpha,\beta)$ of the polynomial. If the set $D^{-1}(0)\subseteq \Omega$ has no interior points, it is suficient to check a single $(\alpha,\beta)$ per connected component of $\Omega\setminus D^{-1}(0)$.

  • $\begingroup$ Thanks for your quick answer. The $a_i$ are indeed continuous (and differentiable). The problem I'm facing is that the discriminant is a very complicated expression. With a numerical computation I can at least verify, that $D(\alpha, \beta) > 0$ for all interior points. $\endgroup$ – Roman Feb 11 '13 at 14:45

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