TY - GEN
T1 - Autonomous First-Order Algebraic q-Difference Equations and Their Rational Solutions
AU - Uncu, Ali
AU - Vo, Thieu
PY - 2026/7/12
Y1 - 2026/7/12
N2 - An autonomous first-order algebraic q-difference equation is a difference equation of the form, where f g K[u, v] is a bivariate polynomial over a field K of characteristic zero, q g Kg{0} is not a root of unity, and σq denotes the q-shift operator. In this paper, we develop an algorithm that decides whether such an equation admits a nonconstant rational solution and, when it does, computes such a solution. The corresponding problem for the additive shift operator has been completely resolved by a reduction to a second-order autonomous difference equation, together with an effective upper bound on polynomial solutions of the latter equation. In contrast, no such upper bound exists in the q-shift setting, and the methods for the additive shift case therefore do not apply. To address this obstacle, we introduce a direct approach grounded in an algebraic-geometric viewpoint.
AB - An autonomous first-order algebraic q-difference equation is a difference equation of the form, where f g K[u, v] is a bivariate polynomial over a field K of characteristic zero, q g Kg{0} is not a root of unity, and σq denotes the q-shift operator. In this paper, we develop an algorithm that decides whether such an equation admits a nonconstant rational solution and, when it does, computes such a solution. The corresponding problem for the additive shift operator has been completely resolved by a reduction to a second-order autonomous difference equation, together with an effective upper bound on polynomial solutions of the latter equation. In contrast, no such upper bound exists in the q-shift setting, and the methods for the additive shift case therefore do not apply. To address this obstacle, we introduce a direct approach grounded in an algebraic-geometric viewpoint.
KW - algebraic curve
KW - algebraic q-difference equation
KW - rational parametrization
KW - rational solution
UR - https://www.scopus.com/pages/publications/105045831484
U2 - 10.1145/3815436.3815470
DO - 10.1145/3815436.3815470
M3 - Chapter in a published conference proceeding
AN - SCOPUS:105045831484
T3 - Proceedings of the International Symposium on Symbolic and Algebraic Computation, ISSAC
SP - 373
EP - 378
BT - ISSAC 2026 - Proceedings of the 2026 International Symposium on Symbolic and Algebraic Computation
A2 - Koutschan, Christoph
A2 - Bostan, Alin
A2 - Pernet, Clement
A2 - Vu, Thi Xuan
PB - Association for Computing Machinery
CY - New York, U. S. A.
T2 - International Symposium on Symbolic and Algebraic Computation, ISSAC 2026
Y2 - 13 July 2026 through 17 July 2026
ER -