The parity problem for reducible cubic forms

An Infinite Family of Cubics with Emergent Reducibility at Depth 1

A polynomial f(x) has emergent reducibility at depth n if f^{\circ k}(x) is irreducible for 0\leq k\leq n-1 but f^{\circ n}(x) is reducible. In this paper we prove that there are infinitely many irreducible cubics f \in \mathbb{Z}[x] with f\circ f reducible by exhibiting a one parameter family with this property.

The divisor problem for binary cubic forms

We investigate the average order of the divisor function at values of totally reducible binary cubic forms and discuss some applications.

On the average number of divisors of reducible quadratic polynomials

We give an asymptotic formula for the divisor sum $\sum_{c<n\leq N}τ\left((n-b)(n-c)\right)$ for integers $b<c$ of the same parity. Interestingly, the coefficient of the main term does not depend on the discriminant as long as it is a full square. We also provide effective upper bounds of the average divisor sum for some of the reducible quadratic polynomials considered before, with the same main term as in the asymptotic formula.

The parity problem for irreducible cubic forms

Let f\in \mathbb{Z}[x,y] be an irreducible homogeneous polynomial of degree 3. We show that f(x,y) has an even number of prime factors as often as an odd number of prime factors.

Enumerative Galois theory for cubics and quartics

We show that there are $O_\varepsilon(H^{1.5+\varepsilon})$ monic, cubic polynomials with integer coefficients bounded by $H$ in absolute value whose Galois group is $A_3$. We also show that the order of magnitude for $D_4$ quartics is $H^2 (\log H)^2$, and that the respective counts for $A_4$, $V_4$, $C_4$ are $O(H^{2.91})$, $O(H^2 \log H)$, $O(H^2 \log H)$. Our work establishes that irreducible non-$S_3$ cubic polynomials are less numerous than reducible ones, and similarly in the quartic setting: these are the first two solved cases of a 1936 conjecture made by van der Waerden.

三次序环类群中2-挠元平均个数

The mean number of $2$-torsion elements in the class groups of cubic orders

本文确定了三次序类群中2-挠元的平均个数，并将其推广到任意奇数维情形，给出了可约SL_N(Z)-轨道的渐近计数公式。