Burchnall-Chaundy多项式与Laurent现象
Burchnall-Chaundy polynomials and the Laurent phenomenon
摘要：
Burchnall-Chaundy多项式$P_n(z)$是由微分递推关系$$P_{n+1}'(z)P_{n-1}(z)-P_{n+1}(z)P_{n-1}'(z)=P_n(z)^2$$与$P_{-1}=P_0(z)=1.$确定的。这种递推关系具有多项式的所有解的事实并不明显，类似于Somos序列的完整性和Laurent现象。我们讨论了这种并行更详细，并将其扩展到两个差分方程$$Q_{n+1}(z+1)Q_{n-1}(z)-Q_{n+1}(z)Q_{n-1}(z+1)=Q_n(z)Q_n(z+1)$$和$$R_{n+1}(z+1)R_{n-1}(z-1)-R_{n+1}(z-1)R_{n-1}(z+1)=R^2_n(z)$$有关的两个不同的KdV型减少Hirota-Miwa和道奇森八面体方程。作为推论，我们有一个新的形式的Burchnall-Chaundy多项式的初始数据$P_n(0)$，这是劳伦特。
英文摘要：
The Burchnall-Chaundy polynomials $P_n(z)$ are determined by the differential recurrence relation $$P_{n+1}'(z)P_{n-1}(z)-P_{n+1}(z)P_{n-1}'(z)=P_n(z)^2$$ with $P_{-1}=P_0(z)=1.$ The fact that this recurrence relation has all solutions polynomial is not obvious and is similar to the integrality of Somos sequences and the Laurent phenomenon. We discuss this parallel in more detail and extend it to two difference equations $$Q_{n+1}(z+1)Q_{n-1}(z)-Q_{n+1}(z)Q_{n-1}(z+1)=Q_n(z)Q_n(z+1)$$ and $$R_{n+1}(z+1)R_{n-1}(z-1)-R_{n+1}(z-1)R_{n-1}(z+1)=R^2_n(z)$$ related to two different KdV-type reductions of the Hirota-Miwa and Dodgson octahedral equations. As a corollary we have a new form of the Burchnall-Chaundy polynomials in terms of the initial data $P_n(0)$, which is shown to be Laurent.
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