Conjecture 1.4 — Shorey

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

Let $L\ge2$, $n_1>\cdots>n_{L-1}>0$, and $b_1,\ldots,b_L\in\mathbb Z\setminus\{0\}$, and write $f(X)=\sum_{i=1}^{L-1}b_iX^{n_i}+b_L$ with $H=\max_i|b_i|$. Assume that $f$ has at least two distinct complex roots. A proper vanishing subsum is a nonempty strict subset of the displayed signed terms whose sum equals zero.

Statement recorded in the supplied source.

Is there $C=C(L,H)>0$ such that any integers $m\ge2$, $x$, and $y$ with $|y|>1$ and $y^m=f(x)$ satisfy either $m\le C$ or the expression $y^m-\sum_{i=1}^{L-1}b_i x^{n_i}-b_L$ has a proper vanishing subsum? [S2]

Short English statement

Is the perfect-power exponent bounded in terms only of the number and size of nonzero coefficients, unless cancellation occurs in a proper subset of the equation terms?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 8700003 (AMR-086-0003). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Michel Waldschmidt, Open Diophantine Problems, arXiv:math/0312440v2 (24 January 2004), Conjecture 1.4 and immediately preceding sparse-polynomial hypotheses. https://arxiv.org/abs/math/0312440.