论文标题
立方，四分和五项多项式的阳性条件
Positivity Conditions for Cubic, Quartic and Quintic Polynomials
论文作者
论文摘要
我们为立方多项式提供了必要且充分的条件，使所有正真实都具有阳性。我们确定了立方多项式无负的集合，但并非全部对于所有正真实，并且明确给出了立方多项式零的点。然后，我们重新制定了一个必要和充分的条件，使四分之一的多项式对于所有积极真实物质都不是不负的。由此，我们得出了四分之一多项式的必要和充分条件，使得所有真实物质都是非负和阳性的。我们的病情明确表现出某些系数的范围和作用，并且具有很强的几何含义。在所有真实的非阴性区域的内部中，都有一个附录曲线。附录的判别物为零，在非负区域内部的另一部分为正。通过使用sturm序列，我们为五级多项式提供了必要且充分的条件，以使所有正真实物质为正且无负。我们表明，对于固定程度的多项式高于或等于四的多项式，如果它们没有真正的根，则其判别因子采用相同的符号，仅取决于该程度，除了较低的尺寸较低的附录集较低的附录集，而判别物获得了零。
We present a necessary and sufficient condition for a cubic polynomial to be positive for all positive reals. We identify the set where the cubic polynomial is nonnegative but not all positive for all positive reals, and explicitly give the points where the cubic polynomial attains zero. We then reformulate a necessary and sufficient condition for a quartic polynomial to be nonnegative for all positive reals. From this, we derive a necessary and sufficient condition for a quartic polynomial to be nonnegative and positive for all reals. Our condition explicitly exhibits the scope and role of some coefficients, and has strong geometrical meaning. In the interior of the nonnegativity region for all reals, there is an appendix curve. The discriminant is zero at the appendix, and positive in the other part of the interior of the nonnegativity region. By using the Sturm sequences, we present a necessary and sufficient condition for a quintic polynomial to be positive and nonnegative for all positive reals. We show that for polynomials of a fixed even degree higher than or equal to four, if they have no real roots, then their discriminants take the same sign, which depends upon that degree only, except on an appendix set of dimension lower by two, where the discriminants attain zero.
