AI 中文总结
该研究将$\reals^d$上的广义Wronskian视为$\reals[x^1,\twodots,x^d]$的$N$元李括号，证明迭代Wronskian的最高总次数渐近增长不超过$N$-斐波那契数，且$d=1$、$k$为奇数时可达该边界。
AI 中文摘要
对于$d\boldsymbol{\u2265}1$个变量的多项式代数$\reals[x^1,\twodots,x^d]$，将$\reals^d$上微分阶数为$k\boldsymbol{\u2265}1$的完全广义Wronskian$W_d^k$视为$\boldsymbol{N}=\binom{d+k}{d}$元李括号。取$N$元多项式组，计算其Wronskian并重复使用新生成的多项式以生成更多多项式。问题是：随着括号迭代次数$n$增加，它们的最大总次数增长有多快？此处引入由递推式$F^{(N)}_n=F^{(N)}_{n-1}+\boldsymbol{\twdots}+F^{(N)}_{n-N}\boldsymbol{\u2208}\naturals$定义的$\boldsymbol{N}$-斐波那契数。我们证明，对于任意初始参数选择，最高总次数序列$\boldsymbol{\textsf{D}}^{(N)}_n\boldsymbol{\u2265}0$的渐近增长（若增长）不超过第$n$个$N$-斐波那契数：$\boldsymbol{\text{lim}}_{n\boldsymbol{\u2192}+\boldsymbol{\u221e}}(\textsf{D}^{(N)}_n/F^{(N)}_n)<\boldsymbol{\u221e}$。我们表明，当$d=1$且$k$为奇数时，最高多项式次数达到$N$-斐波那契数的边界。
英文摘要
For the algebra $\mathbb{R}[x^1,\ldots,x^d]$ of polynomials in $d\geqslant 1$ variables, regard the complete generalised Wronskian $W_d^k$ of differential order $k\geqslant 1$ over $\mathbb{R}^d$ as the $N=\tbinom{d+k}{d}$-ary Lie bracket. Take an $N$-tuple of polynomials, calculate their Wronskian, and keep re-using the newly-created polynomials to produce more of them. The problem is: how fast do their maximal total degrees grow with the number $n$ of iterations of the bracket? Here enter the $N$-bonacci numbers defined by the recurrence $F^{(N)}_n=F^{(N)}_{n-1}+\cdots+F^{(N)}_{n-N}\in \mathbb{N}$. We prove that for any choice of the initial arguments, the sequence of highest total degrees ${\mathsf{D}}^{(N)}_n \geqslant 0$ grows (if at all) asymptotically no faster than the $n$th $N$-bonacci number: $\lim_{n\to+\infty} ({\mathsf{D}}^{(N)}_n/F^{(N)}_n )<\infty$. We show that for $d=1$ and $k$ odd, the highest polynomial degrees do attain the $N$-bonacci bound. Keywords: Differential polynomial, $N$-ary Lie bracket, multivariate Wronskian determinant, Fibonacci numbers, $N$-bonacci numbers, asymptotic growth rate, growth of polynomial degrees, Skolem--Pisot problem.
CommentsNotation and conventions shared with arXiv:2605.27305 [math.RA]; 13 pages, 1 figure; 2 ancillary files attached
