论文标题
5度多项式和笛卡尔的标志规则
Degree 5 polynomials and Descartes' rule of signs
论文作者
论文摘要
对于单变量的真实多项式而没有零系数，笛卡尔的标志规则（通过观察傅立叶的观察完成）表示，其数字$ $ pos $ pos $ pos $ a pos $ a pos $ a pos $ a pos $ n of负根（用多重性计数）由数字$ c $ $ c $和$ c $和$ p $的符号变化和标志保存在$上，以及$ $ $ $ $ nistery $ nistery $ cyeff in nistern $ ccc，即数字。对于5度多项式，A。Albouy和Y.对于高度5，当领先系数为正时，这些都是正面和负根数（所有不同）的数量，以及与笛卡尔符号规则兼容的系数的符号，但对于没有这样的多项式。我们通过图片通过图片来解释这种不存在的以及在所有其他情况下的存在，其中显示了多项式家庭的判别集$ x^5+x^4+ax^3+bx^2+bx^2+cx+d $以及坐标轴。
For a univariate real polynomial without zero coefficients, Descartes' rule of signs (completed by an observation of Fourier) says that its numbers $pos$ of positive and $neg$ of negative roots (counted with multiplicity) are majorized respectively by the numbers $c$ and $p$ of sign changes and sign preservartions in the sequence of its coefficients, and that the differences $c-pos$ and $p-neg$ are even numbers. For degree 5 polynomials, it has been proved by A.~Albouy and Y.~Fu that there exist no such polynomials having three distinct positive and no negative roots and whose signs of the coefficients are $(+,+,-,+,-,-)$ (or having three distinct negative and no positive roots and whose signs of the coefficients are $(+,-,-,-,-,+)$). For degree 5 and when the leading coefficient is positive, these are all cases of numbers of positive and negative roots (all distinct) and signs of the coefficients which are compatible with Descartes' rule of signs, but for which there exist no such polynomials. We explain this non-existence and the existence in all other cases with $d=5$ by means of pictures showing the discriminant set of the family of polynomials $x^5+x^4+ax^3+bx^2+cx+d$ together with the coordinate axes.
