On Nonnegativity of the Generalized Discriminants of Quadratics

Abstract

Discriminants of quadratics have been recently generalized as ${\Delta }_k=k(n-k)c_{n-k}^2-(k+1)(n-k+1)c_{n-k-1}{c}_{n-k+1}$ for a polynomial $f(x)=c_n{x}^n+\cdots +c_{1}x+c_0$ of degree $n\ge 2$ for $1\le k\le n-1$ and it has been shown that ${\Delta }_1\ge 0$ if f has real roots only, [1]. In this article we extend this result to ${\Delta }_k$. Namely, we show that ${\Delta }_k\ge 0$ if f has real roots only for $1<k\le n-1$. As an application, we also demonstrate how to graph a plane quartic without using any calculus tools other than continuity.

MSC:

• 26C10

ACKNOWLEDGMENTS

The author wishes to thank the anonymous referee for helpful suggestions and numerous corrections. This research is partially supported by the Research Foundation of The City University of New York.

DISCLOSURE STATEMENT

No potential conflict of interest was reported by the author.

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Yusuf Z. Gürtaş

YUSUF Z. GÜRTAŞ received his Ph.D. in mathematics from the University of California, Irvine, in 2003. He held visiting positions at various universities before joining the faculty at QCC, CUNY in 2009. He is married and has two sons who don’t hate math.