We consider polynomials of degree $d$ with only real roots and a fixed value of discriminant, and study the problem of minimizing the absolute value of such polynomials at a fixed point off the real line. There are two explicit families of polynomials that turn out to be extremal in terms of this problem. The first family has a particularly simple expression as a linear combination of $d$th powers of two linear functions. Moreover, if the value of the discriminant is not too small, then the roots of the extremal polynomial and the smallest absolute value in question can be found explicitly. The second family is related to generalized Jacobi (or Gegenbauer) polynomials, which helps us to find the associated discriminants. We also investigate the dual problem of maximizing the value of discriminant, while keeping the absolute value of polynomials at a point away from the real line fixed. Our results are then applied to problems on the largest disks contained in lemniscates, and to the minimum energy problems for discrete charges on the real line.

我们考虑度的多项式 $d$仅具有实数根和一个判别式的固定值，并研究使此类多项式的绝对值最小化的问题。在这个问题上，有两个明确的多项式族证明是极好的。第一族具有特别简单的表达形式，即$d$两个线性函数的三次方。此外，如果判别式的值不太小，则可以明确找到极值多项式的根和所讨论的最小绝对值。第二族与广义Jacobi（或Gegenbauer）多项式有关，这有助于我们找到相关的判别式。我们还研究了最大化判别式值，同时将多项式的绝对值保持在远离实线的点上的对偶问题。然后将我们的结果应用于定理线包含的最大磁盘上的问题，以及实线上离散电荷的最小能量问题。