Conjecture 1.5

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 $k\ge2$ and let $\alpha_1,\ldots,\alpha_k\ne0$ lie in a characteristic-zero field $K$, with $\alpha_i/\alpha_j$ not a root of unity for $i\ne j$. For $x_1,\ldots,x_k\in\mathbb Z$, put $M(x)=(\alpha_j^{x_i})_{1\le i,j\le k}$; negative exponents use inverses in $K$.

Statement recorded in the supplied source.

Are there only finitely many $(x_2,\ldots,x_k)\in\mathbb Z^{k-1}$ such that $\det M(0,x_2,\ldots,x_k)=0$, every matrix formed by deleting one row has rank $k-1$, and every matrix formed by deleting one column has rank $k-1$? [S2]

Short English statement

Does the nondegenerate exponential determinant equation with first exponent zero have only finitely many integer solutions?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 8700004 (AMR-086-0004). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Michel Waldschmidt, Open Diophantine Problems, arXiv:math/0312440v2 (24 January 2004), Conjecture 1.5. https://arxiv.org/abs/math/0312440.