Squareful numbers in hyperplanes

Abstract

Let n 4. In this article, we will determine the asymptotic behavior of the size of the set of integral points (a 0 : · · · : a n ) on the hyperplane n i=0 X i = 0 in ‫ސ‬ n such that a i is squareful (an integer a is called squareful if the exponent of each prime divisor of a is at least two) and |a i | B for each i ∈ {0, . . . , n}, when B goes to infinity. For this, we will use the classical Hardy-Littlewood method. The result obtained supports a possible generalization of the Batyrev-Manin program to Fano orbifolds.

Arithmetic of higher-dimensional orbifolds and a mixed Waring problem

“…In the special case m 0 = • • • = m n = 2, work of Van Valckenborgh [14] establishes an asymptotic formula for N (P n−1 , ; B) for all n ≥ 4, which agrees with (1.2). Drawing inspiration from this, we have the following generalisation, which is also in accordance with (1.2).…”

Campana points of bounded height on vector group compactifications

“…The notion of "orbifold rational point" is explored in Campana's papers [18, § 9], [19, § 4], [20, § 12] [71], followed immediately by [16] and more recently by [17]. Work of Schindler and the first author [56] investigates the distribution of Campana points on toric varieties.…”

“…Remark So far few results on the arithmetic of (weak) Campana points are available. Work on points of bounded height goes back to [71], followed immediately by [16] and more recently by [17]. Work of Schindler and the first author [56] investigates the distribution of Campana points on toric varieties.…”

“…To date, Manin‐type problems for sets of Campana points have not been well studied. The only results we are aware of are to be found in [16, 17, 71] and we believe that this research direction is relatively new.…”

Campana points and powerful values of norm forms

“…These two notions have been termed Campana points and weak Campana points in the recent paper [27] of Pieropan, Smeets, Tanimoto and Várilly-Alvarado, in which the authors initiate a systematic quantitative study of points of the former type on smooth Campana orbifolds and prove a logarithmic version of Manin's conjecture for Campana points on vector group compactifications. The only other quantitative results in the literature are found in [5,6,26,32,33], and the former four of these indicate the close relationship between Campana points and m-full solutions of equations. We recall that, given m ∈ Z ≥2 , we say that n ∈ Z\{0} is m-full if all primes in the prime decomposition of n have multiplicity at least m.…”