Weakly Divisible Rings

2024/10/16 by Gaurav Digambar Patil, Patil, Gaurav Digambar
Mathematics · #Rings, Modules, and Algebras #Advanced Topics in Algebra #Algebraic structures and combinatorial models

paper · pdf · doi:10.48550/arxiv.2410.12970

Abstract

We define a new class of rings parameterized by binary forms of a certain type, and give an effective lower bound for the number of such rings whose discriminant is less than a bound X. We also obtain a lower bound for the number of number fields whose ring of integers lies in the above class and whose discriminant is less than a bound X. Our results improve an estimate of Bhargava-Shankar-Wang in \citebhargava2022squarefree. In particular we show the following: \bullet When n≥ 4, the number of rings of rank n over ℤ with discriminant less than or equal to X is ≫_n X^{(1)/(2)+(1)/(n-(4)/(3))}. \bullet When n≥ 6, the number of number fields of degree n with discriminant less than X is ≫_n,ε X^{(1)/(2) +(1)/(n-1) + ((n-3)r_n)/((n-2)(n-1))-ε} where r_n=(η_n)/(n²-4n+3-2η_n (n+(2)/(n-2))) and where η_n is (1)/(5n) if n is odd and is (1)/(88n⁶) when n is even.

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