arXiv subjects
Publications and source records attributed to Gaurav Digambar Patil.
Ekedhal Sieve can be used to count squarefree bounded values of a polynomial closely related to the discriminant polynomial and one can also get a weighted lower bounds for the number of number fields with bounded discriminant which show a limsup much larger than current known bounds.
We establish a Fundamental Theorem of Orders (FTO), which allows us to express any order (in a number field) uniquely as an intersection of \textit{irreducible orders}. Along this decomposition, the index (in the ring of integers) distributes multiplicatively, and the conductor factors into pairwise co-prime ideals. We use it to show a more general version of Furtwangler criterion about the structure of conductors of orders over $\Z$, as this answers a wide variations of such questions. In a future work, we will also give applications to weighted enumeration of number fields.
The Ekedahl sieve is a powerful tool for enumerating arithmetic objects, but traditional formulations relying on inductive steps often yield suboptimal bounds when applied to highly skew boxes. This limitation is particularly restrictive when introducing large modular conditions that compete with the tail-end variables of a binary form. In this paper, we develop a specialized variant of the Ekedahl sieve tailored to the singular locus of the discriminant polynomial. By exploiting the specific non-degeneracy properties of the discriminant outside its two most extreme coefficients, we bypass the standard inductive framework, reducing the sieve to a highly efficient two-step process in some cases, and a one step process in others. We establish robust generic tail-end estimates as well as squarefree, power-saving bounds that seamlessly incorporate external modular conditions. This optimization maximizes the permissible range of the sieving modulus, yielding improved error terms for the enumeration of bounded squarefree values of certain polynomials and providing the foundational geometric sieve estimates required for the weighted enumeration of number fields by discriminant.
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 \cite{bhargava2022squarefree}. In particular we show the following: $\bullet$ When $n\ge 4,$ the number of rings of rank $n$ over $\mathbb{Z}$ with discriminant less than or equal to $X$ is $$\gg_n X^{\frac{1}{2}+\frac{1}{n-\frac{4}{3}}}.$$ $\bullet$ When $n\ge 6,$ the number of number fields of degree $n$ with discriminant less than $X$ is $$\gg_{n,ε} X^{\frac{1}{2} +\frac{1}{n-1} + \frac{(n-3)r_n}{(n-2)(n-1)}-ε}$$ where $r_n=\frac{η_n}{n^2-4n+3-2η_n (n+\frac{2}{n-2})}$ and where $η_n$ is $\frac{1}{5n}$ if $n$ is odd and is $\frac{1}{88n^6}$ when $n$ is even.