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

Abstract

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.