Conjecture 4.13 — Laurent-Roy

Definitions and mathematical statement

Let $m\geq1$ be an integer and $\theta\in{\mathbb{C}}^m$. The Laurent--Roy conjecture in the source asserts that there exists $c=c(\theta,m)>0$ such that, for every integer $k$ with $0\leq k\leq m$, infinitely many integers $T\geq1$ admit a ${\mathbb{Q}}$-defined algebraic set $Z\subset{\mathbb{C}}^m$ of dimension exactly $k$ and a point $\alpha\in Z$ satisfying \[ t(Z)\leq T^{m-k},\qquad |\theta-\alpha|\leq\exp\{-cT^{m+1}\}. \] The constant is chosen before $k$ and is independent of $T,Z,\alpha$. The infinitely many $T$ may depend on $k$; the source does not require one common infinite set of $T$ working for every $k$, nor every sufficiently large $T$.

An algebraic set defined over ${\mathbb{Q}}$ means a common zero locus of polynomials in ${\mathbb{Q}}[x_1,\ldots,x_m]$, with its usual algebraic dimension. No irreducibility or equidimensionality assumption is supplied. The point $\alpha$ is an arbitrary complex point on $Z$, not stated to be algebraic over ${\mathbb{Q}}$ in positive dimension. The norm $|\cdot|$ is unspecified; a fixed Euclidean norm is a conventional interpretation requiring source confirmation.

The crucial size $t(Z)$ is not defined in the record. It must be recovered, including its degree/height and affine/projective normalizations if applicable; none is guessed. This affects the endpoint $k=m$, where the required bound is $t(Z)\leq1$. Both endpoints $k=0,m$ are retained. The summary's mention of hypersurfaces does not reduce the source's full range of dimensions to $k=m-1$.

Short English statement

Approximate a fixed complex vector exponentially closely by a point on a rationally defined algebraic set of each prescribed dimension, with the exact size bound $t(Z)\leq T^{m-k}$. The size invariant is missing, but the uniform constant and infinitely-many-parameter quantifiers are explicit.

Sources

• Local UnsolvedMath registry, permanent ID 8700055, exact source code catalogue entry, "Conjecture 4.13---Laurent--Roy"; registry attribution Michel Waldschmidt, no proposed year supplied. Registry status open, classification OPEN-TRIAGE.
• Michel Waldschmidt, Open Diophantine Problems (2004), Conjecture 4.13, https://arxiv.org/abs/math/0312440. Supplied source-TeX extraction accessed July 29, 2026.
• The supplied summary attributes the conjecture to Laurent--Roy and mentions related weaker/transposed approximation bounds by Roy and Laurent--Roy without precise references. It reports the exact sharp statement open in its accessed literature; this is not an independent current-status check.
• Provenance flags current primary-literature status and redistribution terms NEEDS_REVIEW; the historical survey presents numbered open problems and conjectures.