Squarefree values of polynomial discriminants I (1611.09806v3)

Abstract: We determine the density of monic integer polynomials of given degree $n>1$ that have squarefree discriminant; in particular, we prove for the first time that the lower density of such polynomials is positive. Similarly, we prove that the density of monic integer polynomials $f(x)$, such that $f(x)$ is irreducible and $\mathbb Z[x]/(f(x))$ is the ring of integers in its fraction field, is positive, and is in fact given by $\zeta(2)^{-1}$. It also follows from our methods that there are $\gg X^{1/2+1/n}$ monogenic number fields of degree $n$ having associated Galois group $S_n$ and absolute discriminant less than $X$, and we conjecture that the exponent in this lower bound is optimal.

Squarefree Values of Polynomial Discriminants I: An Overview

The paper by Bhargava, Shankar, and Wang addresses the density of monic integer polynomials with squarefree discriminants and extends the understanding of number field monogenicity. It articulates the theoretical framework underpinning polynomial discriminants and illustrates the implications for algebraic number theory, particularly in relation to monogenic number fields and their properties.

Main Results

1. Density of Monic Polynomials: The authors determine that the density of monic integer polynomials of a given degree $n > 1$ with a squarefree discriminant is positive. Their methodology involves ordering by height $H(f) = \max \{ |a_i| / i \}$, demonstrating that as $n → ∞$, the probability of a polynomial being squarefree approaches approximately $30.7056\%$.
2. Ring of Integers: A significant contribution is proving that $\approx 60.7927\%$ of monic integer polynomials $f(x)$ ensure $\mathbb{Z}[x]/(f(x))$ is the maximal order in its fraction field, aligning this density with $\zeta(2)^{-1}$. This finding reinforces Hilbert's irreducibility theorem, confirming the ubiquity of irreducible polynomials in defining number fields.
3. Monogenic Number Fields: The paper conjectures optimality in the lower bound count of monogenic number fields and enhances existing bounds by prior researchers. Specifically, it establishes that the number of isomorphism classes of monogenic $S_n$-number fields of degree $n$ grows as $\gg X^{1/2 + 1/n}$.
4. Power-Saving Error Terms: The authors provide power-saving accuracy in their asymptotic estimates, stipulating that error terms are small relative to the main term, which streamlines identifying lower density bounds.

Theoretical Implications

• Polynomial Discriminants: Establishing density bounds provides insights into the algebraic structure and distribution of polynomials, influencing how polynomials can be employed in constructing number fields with predictable properties.
• Monogenic Fields: Showing a large class being monogenic directly connects polynomial formulations with field expansions, essential in applications that demand computational efficiency and structural understanding.

Future Directions

The paper leaves several avenues for further exploration:

• Higher-Degree Polynomial Analysis: Follow-up work (Part II) anticipates extending the results to non-monic polynomials, promising broader applications and further theoretical underpinning for discriminant-related theories and conjectures.
• Local Conditions: The authors note that analogous results can be generalized under finite sets of local conditions, hinting at potential advancement in localization techniques in number theory.
• Distribution of Zeros: Error term analysis applications towards the distribution of low-lying zeros in Dedekind zeta functions of monogenic fields suggest deeper links between polynomial discriminants and analytic number theory.

Conclusion

The paper is thorough in its evaluation of polynomial discriminants, offering quantitative enhancements to the understanding of number theory. It makes substantial progress in affirming the density of polynomials with specific discriminant properties and examines implications on the construction and characterization of number fields. Researchers invested in algebraic number theory could build on these findings to explore further conjectures or develop new applications in computational aspects of number fields.