2018/08/04 by Pavlo Yatsyna, Yatsyna, Pavlo · 2 citations
Computer Science · Mathematics · #11E12 #11H06 #11R09 #11R11 #11R16 #11R18 #Algebraic Geometry and Number Theory #Analytic Number Theory Research #Coding theory and cryptography #FOS: Mathematics #Number Theory (math.NT) #Polynomial and algebraic computation
paper · pdf · doi:10.48550/arxiv.1808.01441
openalex publication_date 2018/08/04 · openalex created_date 2022/08/04 · openalex updated_date 2026/07/28
We show that if K is a monogenic, primitive, totally real number field,\nthat contains units of every signature, then there exists a lower bound for the\nrank of integer universal quadratic forms defined over K. In particular, we\nextend the work of Blomer and Kala, to show that there exist infinitely many\ntotally real cubic number fields that do not have a universal quadratic form of\na given rank defined over them. For the real quadratic number fields with a\nunit of negative norm, we show that the minimal rank of a universal quadratic\nform goes to infinity as the discriminant of the number field grows. These\nresults follow from the study of interlacing polynomials. Specifically, we show\nthat there are only finitely many irreducible monic polynomials related to\nprimitive number fields of a given degree, that have a bounded number of\ninterlacing polynomials.\n