{"language":[{"iso":"eng"}],"type":"preprint","date_updated":"2026-02-09T07:49:17Z","citation":{"chicago":"Arango-Piñeros, Santiago, Fabian Gundlach, Robert J. Lemke Oliver, Kevin J. McGown, Will Sawin, Allechar Serrano López, Arul Shankar, and Ila Varma. “Counting Number Fields of Fixed Degree by Their Smallest Defining Polynomial.” <i>ArXiv:2602.06943</i>, 2026.","bibtex":"@article{Arango-Piñeros_Gundlach_Lemke Oliver_McGown_Sawin_Serrano López_Shankar_Varma_2026, title={Counting number fields of fixed degree by their smallest defining polynomial}, journal={arXiv:2602.06943}, author={Arango-Piñeros, Santiago and Gundlach, Fabian and Lemke Oliver, Robert J. and McGown, Kevin J. and Sawin, Will and Serrano López, Allechar and Shankar, Arul and Varma, Ila}, year={2026} }","short":"S. Arango-Piñeros, F. Gundlach, R.J. Lemke Oliver, K.J. McGown, W. Sawin, A. Serrano López, A. Shankar, I. Varma, ArXiv:2602.06943 (2026).","apa":"Arango-Piñeros, S., Gundlach, F., Lemke Oliver, R. J., McGown, K. J., Sawin, W., Serrano López, A., Shankar, A., &#38; Varma, I. (2026). Counting number fields of fixed degree by their smallest defining polynomial. In <i>arXiv:2602.06943</i>.","ama":"Arango-Piñeros S, Gundlach F, Lemke Oliver RJ, et al. Counting number fields of fixed degree by their smallest defining polynomial. <i>arXiv:260206943</i>. Published online 2026.","ieee":"S. Arango-Piñeros <i>et al.</i>, “Counting number fields of fixed degree by their smallest defining polynomial,” <i>arXiv:2602.06943</i>. 2026.","mla":"Arango-Piñeros, Santiago, et al. “Counting Number Fields of Fixed Degree by Their Smallest Defining Polynomial.” <i>ArXiv:2602.06943</i>, 2026."},"author":[{"first_name":"Santiago","last_name":"Arango-Piñeros","full_name":"Arango-Piñeros, Santiago"},{"full_name":"Gundlach, Fabian","last_name":"Gundlach","first_name":"Fabian","id":"100450"},{"last_name":"Lemke Oliver","first_name":"Robert J.","full_name":"Lemke Oliver, Robert J."},{"full_name":"McGown, Kevin J.","last_name":"McGown","first_name":"Kevin J."},{"last_name":"Sawin","first_name":"Will","full_name":"Sawin, Will"},{"full_name":"Serrano López, Allechar","last_name":"Serrano López","first_name":"Allechar"},{"last_name":"Shankar","first_name":"Arul","full_name":"Shankar, Arul"},{"last_name":"Varma","first_name":"Ila","full_name":"Varma, Ila"}],"title":"Counting number fields of fixed degree by their smallest defining polynomial","status":"public","abstract":[{"text":"When do two irreducible polynomials with integer coefficients\r\n  define the same number field? One can define an action of\r\n  $\\mathrm{GL}_2 \\times \\mathrm{GL}_1$ on the space of polynomials of degree $n$ so that for any two\r\n  polynomials $f$ and $g$ in the same orbit, the roots of $f$ may be expressed\r\n  as rational linear transformations of the roots of $g$; thus, they generate\r\n  the same field. In this article, we show that almost all polynomials of\r\n  degree $n$ with size at most $X$ can only define the same number field as\r\n  another polynomial of degree $n$ with size at most $X$ if they lie in the\r\n  same orbit for this group action. (Here we measure the size of polynomials by\r\n  the greatest absolute value of their coefficients.)\r\n  This improves on work of Bhargava, Shankar, and Wang, who proved a similar\r\n  statement for a positive proportion of polynomials. Using this result, we\r\n  prove that the number of degree $n$ fields such that the smallest polynomial\r\n  defining the field has size at most $X$ is asymptotic to a constant times\r\n  $X^{n+1}$ as long as $n\\geq 3$. For $n = 2$, we obtain a precise asymptotic of\r\n  the form $\\frac{27}{π^2} X^2$.","lang":"eng"}],"date_created":"2026-02-09T07:48:05Z","_id":"64068","external_id":{"arxiv":["2602.06943"]},"publication":"arXiv:2602.06943","year":"2026","user_id":"100450"}