Iterate Wronskians over ${\mathbb{R}}^d$ as $N$-ary brackets on $\mathbb{R}[x^1,\dots ,x^d]$: the $N$-bonacci numbers bound the highest total degrees

Abstract

For the algebra $\mathbb{R}[x^1,\dots ,x^d]$ of polynomials in $d⩾1$ variables, regard the complete generalised Wronskian $W_d^k$ of differential order $k⩾1$ over ${\mathbb{R}}^d$ as the $N=\left(\genfrac{}{}{0ex}{}{d+k}{d}\right)$-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 $F_n^{(N)}=F_{n-1}^{(N)}+\cdots +F_{n-N}^{(N)}\in \mathbb{N}$. We prove that for any choice of the initial arguments, the sequence of highest total degrees ${\mathsf{D}}_n^{(N)}⩾0$ 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.

Notation and conventions shared with arXiv:2605.27305 [math.RA]; 13 pages, 1 figure; 2 ancillary files attached