Conjecture 4.16 — Quantitative Refinement of Schanuel's Conjecture

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 $n,d\ge1$. For an integer polynomial $P$ in $2n$ variables, its degree means total degree and its usual height is the largest absolute coefficient. In $R=\mathbb Q[X_1,\ldots,X_n,Y_1,\ldots,Y_n]$, the rank of a prime ideal is its height (the supremum of lengths of strict chains of prime ideals ending at it); the rank of an ideal $I$ is $\min_{\mathfrak p\supset I}\operatorname{ht}\mathfrak p$.

Statement recorded in the supplied source.

Let $x_1,\ldots,x_n\in\mathbb C$ be linearly independent over $\mathbb Q$ and suppose \[ \forall\varepsilon>0\ \exists H_0>0\ \forall H\ge H_0\ \forall(h_1,\ldots,h_n)\in\mathbb Z^n, \quad 0<\max_i|h_i|\le H\Longrightarrow \left|\sum_i h_ix_i\right|\ge e^{-H^{\varepsilon}}. \] Is there a constant $C=C(x_1,\ldots,x_n,d)>0$ such that, for every integer $H\ge2$ and every $P_1,\ldots,P_{n+1}\in\mathbb Z[X_1,\ldots,X_n,Y_1,\ldots,Y_n]$ of total degrees at most $d$ and heights at most $H$ whose ideal in $R$ has rank $n+1$, \[ \sum_{j=1}^{n+1}|P_j(x_1,\ldots,x_n,e^{x_1},\ldots,e^{x_n})|\ge H^{-C}? \] [S2]

Short English statement

Under the stated quantitative independence hypothesis, must polynomial systems of the specified codimension stay at least polynomially far from vanishing on the exponential tuple?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 8700058 (AMR-086-0058). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Michel Waldschmidt, Open Diophantine Problems, arXiv:math/0312440v2 (24 January 2004), Conjecture 4.16, p.42; ideal-rank definition, p.41. https://arxiv.org/abs/math/0312440v2.