Decidability of zero hitting by a matrix exponential
Let $\overline{\mathbb{Q}}_{\mathbb{R}}$ be the field of real algebraic numbers, each given exactly by an integer polynomial vanishing at it and a rational interval isolating it among the polynomial's roots. For $A \in \overline{\mathbb{Q}}_{\mathbb{R}}^{d \times d}$ and $u, v \in \overline{\mathbb{Q}}_{\mathbb{R}}^{d}$ the function $t \mapsto u^{\mathsf T} e^{tA} v$ solves a homogeneous linear differential equation with constant real algebraic coefficients and real algebraic initial values, and every such solution arises in this way, so the question below is the continuous Skolem problem (Bell, Delvenne, Jungers and Blondel; Chonev, Ouaknine and Worrell). Bell, Delvenne, Jungers and Blondel proved it decidable for $d \le 2$ and $I = [0, \infty)$. Chonev, Ouaknine and Worrell proved it decidable for bounded $I$ subject to Schanuel's conjecture, and showed that an algorithm for unbounded $I$ would make the Diophantine-approximation types of all real algebraic numbers computable; they later proved it decidable for $d \le 8$ and every $I$, again subject to Schanuel's conjecture. The zero sought need not be rational or algebraic, and $A$ need not be diagonalizable.
Problem. Is there a single algorithm which, given $d \ge 1$, $A \in \overline{\mathbb{Q}}_{\mathbb{R}}^{d \times d}$, $u, v \in \overline{\mathbb{Q}}_{\mathbb{R}}^{d}$ and an interval $I$, either $I = [a, b]$ with rational $0 \le a \le b$ or $I = [a, \infty)$ with rational $a \ge 0$, always halts and decides whether $u^{\mathsf T} e^{tA} v = 0$ for some real $t \in I$?
Source: Bell, Delvenne, Jungers and Blondel (2010), on [0, ∞), with the bounded case as Open Problem 17; for intervals with rational endpoints, Chonev, Ouaknine and Worrell (2016).
Imported from github.com/ajt60gaibb/OpenProblemsInNLA.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- NT — Number Theory
- Secondary
- LO, NA
- Keywords
- continuous Skolem problem, decidability, exponential polynomials, OpenProblemsInNLA
- Literature checked
- Oct. 7, 2026 by clemens-admin
References
@article{bell2010,
eprint = {0809.2189},
archivePrefix = {arXiv},
author = {Bell, Paul C. and Delvenne, Jean-Charles and Jungers, Rapha{\"e}l M. and Blondel, Vincent D.},
title = {The continuous {S}kolem--{P}isot problem},
journal = {Theoret. Comput. Sci.},
volume = {411},
number = {40--42},
year = {2010},
pages = {3625--3634},
doi = {10.1016/j.tcs.2010.06.005},
}
@inproceedings{chonev2016,
author = {Chonev, Ventsislav and Ouaknine, Jo{\"e}l and Worrell, James},
title = {On the {S}kolem Problem for Continuous Linear Dynamical Systems},
booktitle = {43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)},
series = {LIPIcs},
volume = {55},
publisher = {Schloss Dagstuhl--Leibniz-Zentrum f{\"u}r Informatik},
year = {2016},
pages = {100:1--100:13},
doi = {10.4230/LIPIcs.ICALP.2016.100},
eprint = {1506.00695},
archivePrefix = {arXiv},
}
@article{chonev2023,
author = {Chonev, Ventsislav and Ouaknine, Jo{\"e}l and Worrell, James},
title = {On the Zeros of Exponential Polynomials},
journal = {J. ACM},
volume = {70},
number = {4},
year = {2023},
pages = {26:1--26:26},
doi = {10.1145/3603543},
}
@misc{openproblemsinnla-mf20,
author = {Townsend, Alex},
title = {{MF-20}: Decidability of zero hitting by a matrix exponential},
howpublished = {Open Problems in Numerical Linear Algebra, GitHub repository},
year = {2026},
url = {https://github.com/ajt60gaibb/OpenProblemsInNLA/blob/main/matrix-functions-and-stability/MF-20/README.md},
note = {Read at commit b8c7852, 2026-10-07},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00168, "Decidability of zero hitting by a matrix exponential", clopen.solutions.