{"paper":{"title":"The existence of small defining polynomials for number fields, II","arxiv_id":"2611.09031","primary_cat":"math.NT","submitted_at":"2026-11-06T06:05:18+00:00","version_compiled":"v1"},"notice":"Every quoted sentence is the authors' text from the arXiv listing abstract. Labels and probabilities are the outputs of classifier-1.13.0 on the question bank abstract-p1-20260923, exactly as returned. No sentence in this record was written by a language model.","schema":"pith.compiled-record/v0","status":"compiled","abstract":{"source":"arXiv listing abstract v1","sentences":[{"i":0,"p":0.91,"id":"s0","own":true,"role":"result","text":"We prove that every number field of odd degree $n\\geq 3$ and absolute discriminant $D$ has an integral defining polynomial whose coefficients are $O_n(D^{(n-1)/(2n-1)})$ in size.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.96,"main_claim_p":0.91,"quantitative":true,"quantitative_p":0.98,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.91,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.09,"interpretation":0.0}},{"i":1,"p":0.4,"id":"s1","own":false,"role":"interpretation","text":"This strengthens, in odd-degree cases, the recently established bound of $O_n(D^{1/2})$ for all degrees $n$, which was conjectured in 1998 by Ruppert and is now known to be optimal in even degrees.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.16,"main_claim_p":0.12,"quantitative":true,"quantitative_p":0.99,"role_probabilities":{"gap":0.01,"method":0.0,"result":0.1,"background":0.35,"limitation":0.0,"future_work":0.0,"contribution":0.14,"interpretation":0.4}},{"i":2,"p":0.93,"id":"s2","own":true,"role":"result","text":"More generally, we prove that this strengthened bound holds for all degree-$n$ fields having a real embedding and no subfield of index two.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.94,"main_claim_p":0.12,"quantitative":true,"quantitative_p":0.58,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.93,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.07,"interpretation":0.0}},{"i":3,"p":1.0,"id":"s3","own":true,"role":"method","text":"The proof makes use of the lattice shapes of the ring of integers and its trace dual, the inverse different.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.76,"main_claim_p":0.02,"quantitative":false,"quantitative_p":0.04,"role_probabilities":{"gap":0.0,"method":1.0,"result":0.0,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.0,"interpretation":0.0}},{"i":4,"p":0.88,"id":"s4","own":true,"role":"result","text":"Assuming the quasi-Riemann hypothesis for Dedekind zeta functions, as recently announced by Cureton and Thorner, we obtain an exponent smaller than $1/2$ for every field with no index-two subfield, of any signature.","hedge":{"p":0.7,"label":"asserted"},"own_p":0.81,"main_claim_p":0.05,"quantitative":true,"quantitative_p":0.96,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.88,"background":0.0,"limitation":0.04,"future_work":0.0,"contribution":0.08,"interpretation":0.0}},{"i":5,"p":1.0,"id":"s5","own":true,"role":"result","text":"In the case of primitive fields, we obtain the exponent $(n-1)/(2n)$.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.93,"main_claim_p":0.07,"quantitative":true,"quantitative_p":0.96,"role_probabilities":{"gap":0.0,"method":0.0,"result":1.0,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.0,"interpretation":0.0}},{"i":6,"p":0.99,"id":"s6","own":true,"role":"result","text":"In particular, we prove that the optimal exponent for cubic fields is $1/3$.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.94,"main_claim_p":0.13,"quantitative":true,"quantitative_p":0.98,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.99,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.0,"interpretation":0.01}}]},"compiled":{"main_claim":{"p":0.99,"text":"We prove that every number field of odd degree $n\\geq 3$ and absolute discriminant $D$ has an integral defining polynomial whose coefficients are $O_n(D^{(n-1)/(2n-1)})$ in size.","location":"abstract 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