A small value estimate in dimension two involving translations by rational points (Q2810674)

The authors show that ``if a sequence of non-zero polynomials in \(\mathbb{Z}[X_1,X_2]\) takes small values at translates pf a fixed point \((\xi, \eta)\) by multiples of a fixed rational point within the group \(\mathbb{C}\times\mathbb{C}^*\), then \(\xi\) and \(\eta\) are both algebraic over \(\mathbb{Q}\).''NEWLINENEWLINEIn particular, for a polynomial \(P\in\mathbb{Z}[X_1,X_2]\) let \(\|P\|\) denote the largest absolute value of its coefficients, and for a real number \(x\) let \(\lfloor x\rfloor\) denote the integer part of \(x\). The authors prove the following theorem.NEWLINENEWLINETheorem. Let \((\xi,\eta)\in\mathbb{C}\times\mathbb{C}^*\) and \((r,s)\in\mathbb{Q}\times\mathbb{Q}^*\) with \((r,s)\neq (0,\pm 1)\), and let \(\sigma,\beta,\) and \(\nu\) be real numbers satisfying NEWLINE\[NEWLINE1\leq \sigma<2,\quad \beta>\sigma+1,\quad \nu>\begin{cases} 2+\beta-\sigma & \text{if }\sigma\geq 3/2\\ 2+\beta-\sigma +\frac{(\sigma-1)(3-2\sigma)}{2+\beta-2\sigma} & \text{if }\sigma< 3/2.\end{cases}NEWLINE\]NEWLINE Suppose further that for each sufficiently large positive integer \(D\), there exists a non-zero polynomial \(P_D\in\mathbb{Z}[X_1,X_2]\) such that NEWLINE\[NEWLINE\deg P_D\leq D,\quad \|P_D\|\leq e^{D^\beta},\quad \max_{0\leq i<4\lfloor D^\sigma\rfloor} |P_D(\xi+ir,\eta s^i)|\leq e^{-D^\nu}.NEWLINE\]NEWLINE Then \(\xi\) and \(\eta\) are algebraic over \(\mathbb{Q}\).NEWLINENEWLINENote that the constraint \(\sigma \geq 3/2\) is best possible.NEWLINENEWLINEThe proof mainly follows the method of \textit{D.~Roy} [Mathematika 59, No. 2, 333--363 (2013; Zbl 1335.11060)], though there are a few new ingredients as well, including an improvement of a formula of \textit{K.~Mahler} [Acta Math.~Acad.~Sci.~Hungar. 18, 83--96 (1967; Zbl 0207.35602)].