We show that if $K$ is a monogenic, primitive, totally real number field, that contains units of every signature, then there exists a lower bound for the rank of integer universal quadratic forms defined over $K$. In particular, we extend the work of Blomer and Kala, to show that there exist infinitely many totally real cubic number fields that do not have a universal quadratic form of a given rank defined over them. For the real quadratic number fields with a unit of negative norm, we show that the minimal rank of a universal quadratic form goes to infinity as the discriminant of the number field grows. These results follow from the study of interlacing polynomials. Specifically, we show that there are only finitely many irreducible monic polynomials related to primitive number fields of a given degree, that have a bounded number of interlacing polynomials.

中文翻译：

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全实数域中具有整数系数的通用二次型的秩的下界

我们证明，如果 $K$ 是一个单基因的、原始的、完全实数的域，它包含每个签名的单位，那么在 $K$ 上定义的整数通用二次型的秩存在一个下界。特别是，我们扩展了 Blomer 和 Kala 的工作，以表明存在无限多个完全实数的立方数域，它们没有定义在它们上的给定秩的通用二次形式。对于具有负范数单位的实二次数域，我们表明，随着数域的判别式的增长，通用二次型的最小秩趋于无穷大。这些结果来自于对隔行多项式的研究。具体来说，我们证明只有有限多个不可约的单调多项式与给定度数的原始数域相关，