你一直想知道的关于实多项式的事情（但害怕问）
Something You Always Wanted to Know About Real Polynomials (But Were Afraid to Ask)
摘要：
著名的笛卡尔规则的迹象，从1637年给出了上界的数目正根的一个真正的一元多项式方面的一些符号变化的系数，一直是一个不可或缺的来源的灵感，为一代又一代的数学家。试图扩展和锐化这一规则，我们考虑下面的一组所有真正的一元多项式的一个给定的程度，一个给定的收集的迹象，他们的系数，和一个给定的数量的积极和消极的根源。尽管我们研究的主要对象的基本定义，这是一个非平凡的问题，对于相应的集合是非空的符号模式和正负根的数量。本文的主要结果是发现了一个新的无穷族符号模式和正负根个数的不可实现组合。
英文摘要：
The famous Descartes' rule of signs from 1637 giving an upper bound on the number of positive roots of a real univariate polynomials in terms of the number of sign changes of its coefficients, has been an indispensable source of inspiration for generations of mathematicians. Trying to extend and sharpen this rule, we consider below the set of all real univariate polynomials of a given degree, a given collection of signs of their coefficients, and a given number of positive and negative roots. In spite of the elementary definition of the main object of our study, it is a non-trivial question for which sign patterns and numbers of positive and negative roots the corresponding set is non-empty. The main result of the present paper is a discovery of a new infinite family of non-realizable combinations of sign patterns and the numbers of positive and negative roots.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
