On Positivity and Minimality for Second-Order Holonomic Sequences
On Positivity and Minimality for Second-Order Holonomic Sequences
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关于二阶完整序列的正性和极小性
DOI：
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发表时间：
2020
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通讯作者：
J. Worrell
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作者：
George Kenison;O. Klurman;Engel Lefaucheux;F. Luca;P. Moree;Joël Ouaknine;Markus A. Whiteland;J. Worrell
无限序列 $langle{u_n}
如果实数的angle_{ninmathbb{N}}$满足与多项式系数的线性递推关系，则它是完整的（也称为P-递归或P-有限）。如果每个 $u_n geq 0$，这样的序列被称为正数，如果给定任何其他线性独立序列 $langle{v_n}，则称为最小数
mathbb{N}}$中的angle_{n满足相同的递推关系，比率$u_n/v_n$收敛到$0$。在本文中，我们关注满足二阶递推 $g_3(n)u_n = g_2(n)u_{n-1} + g_1(n)u_{n-2}$ 的完整序列，其中 mathbb{Q}[n]$ 中的每个系数 $g_3, g_2,g_1 是次数最多为 $1$ 的多项式。我们得出两个主要结果。首先，我们表明，决定此类序列的积极性可以简化为决定最小性。其次，我们证明确定极小值相当于确定某些数值表达式（称为周期、指数周期和类周期积分）是否等于零。周期和相关表达式是代数几何和数论中的经典研究对象，一些已建立的猜想（特别是 Kontsevich 和 Zagier 的猜想）暗示它们存在可判定的等式问题，这反过来又需要一大类二阶完整序列的正性和极小性的可判定性。
An infinite sequence $langle{u_n}
angle_{ninmathbb{N}}$ of real numbers is holonomic (also known as P-recursive or P-finite) if it satisfies a linear recurrence relation with polynomial coefficients. Such a sequence is said to be positive if each $u_n geq 0$, and minimal if, given any other linearly independent sequence $langle{v_n}
angle_{n inmathbb{N}}$ satisfying the same recurrence relation, the ratio $u_n/v_n$ converges to $0$. In this paper, we focus on holonomic sequences satisfying a second-order recurrence $g_3(n)u_n = g_2(n)u_{n-1} + g_1(n)u_{n-2}$, where each coefficient $g_3, g_2,g_1 in mathbb{Q}[n]$ is a polynomial of degree at most $1$. We establish two main results. First, we show that deciding positivity for such sequences reduces to deciding minimality. And second, we prove that deciding minimality is equivalent to determining whether certain numerical expressions (known as periods, exponential periods, and period-like integrals) are equal to zero. Periods and related expressions are classical objects of study in algebraic geometry and number theory, and several established conjectures (notably those of Kontsevich and Zagier) imply that they have a decidable equality problem, which in turn would entail decidability of Positivity and Minimality for a large class of second-order holonomic sequences.