Probing the generalization of the abc conjecture to more than 3 variables

Question

Browkin and Brzezinski, in "Some remarks on the $abc$-conjecture", Math. Comp. 62 (1994), no. 206, 931–939, state the following generalization of the $abc$ conjecture to more than three variables:

Given $n\ge3$, let $a_1,\dots,a_n \in \Bbb Z$ satisfy $\gcd(a_1,a_2,\dots,a_n)=1$ and $a_1+\dots+a_n=0$, while no proper subsum of the $a_j$ equals $0$. Then for every $\varepsilon>0$, $$ \max\{|a_1|,\dots,|a_n|\} \ll_{n,\varepsilon} R(|a_1\cdots a_n|)^{2n-5+\varepsilon}, $$ where $R(m)$ denotes the radical of $m$ (the product of the distinct primes dividing $m$).

I have two questions about this conjecture.

First, the authors give constructions of $n$-tuples that show that the exponent $2n-5$ on the right-hand side cannot be improved. For example, when $n=4$, one takes any $abc$ triple $(a,b,c)$ and chooses $(a_1,a_2,a_3,a_4)=(a^3,b^3,3abc,-c^3)$, so that $$ R(|a_1a_2a_3a_4|)^{2\cdot4-5} \le (3R(abc))^3 \le 27c^3 = 27\max\{|a_1|,|a_2|,|a_3|,|a_4|\}. $$ On the other hand, a probabilistic argument suggests (I believe) that the exponent $1+\varepsilon$ should suffice. Is the exponent $2n-5$ present only because of integer points on certain lower-dimension varieties like $y^3=-27wxz$, on which the examples $(a^3,b^3,3abc,-c^3)$ all live?

Second, note that the $4$-tuples given above are only relatively prime, not pairwise relatively prime. Has anyone mulled over whether the exponent $2n-5$ can be reduced if one simply strengthens the hypothesis to pairwise coprimality?

(An earlier version of this question wondered why the "no vanishing subsums" condition was present, but that has been answered to my satisfaction.)

Answer 1

In A more general abc conjecture, p. 7 Paul Vojta conjectures exponent $1 + \epsilon$ outside a proper Zariski-closed subset

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Your question appears in The abc-conjecture and the n-conjecture,Coen Ramaekers p. 23

5.1 Strong $n$-conjecture. Adding pairwise coprimality, the exponent is $1$ without additional restrictions. Searching the web for it returns only 3 hits.

Answer 2

Vojta's conjecture predicts $1+\epsilon$ for $n$-tuples outside a closed subvariety.

I think the case of subsums vanishing can lead to things like $a_i+b_i+c_i=0,i=1,2$ and $sa_1+sb_1+sc_1+ta_2+tb_2+tc_2=0$ with $(s,t)=1$ and $s,t$ as big as you like causing trouble.

Answer 3

The relative primality certainly plays an important role. One can formulate a conjecture that accommodates the extent to which a particular summand is pairwise coprime, 3-wise coprime, etc, with other elements. My memory is that the paper of Brownawell and Masser:

Vanishing sums in function fields. Math. Proc. Cambridge Philos. Soc. 100 (1986), no. 3, 427–434.

provides a good function field analogue from which to formulate such a conjecture, though I do not remember the output (except that if one rules out vanishing subsums ... as plainly one should ... then for $n$-wise coprime $n$-tuples the appropriate exponent from such work is $(n-1)(n-2)/2$). Felipe has a paper at the same time (maybe preceded BM?) that likely is just as useful in this regard.

Another "special subvariety" example is:

$$x^4+y^4+(x+y)^4=2(x^2+xy+y^2)^2,$$

where if one takes $x+y+z=0$ as an extremal $abc$-example, then the summands can be made almost pairwise coprime, and the relevant $abcd$-exponent is $4/3$. I think that I first heard about this example from Andrew Granville in the early 1990's.