33

You should refer to Lemma 10 (page-233) in this paper by Schinzel where he proves that for any polynomial $F(x)$ of degree $d$ we have a polynomial $G(x)$ of degree $d-1$ such that their composition is reducible.


15

Yes, it is true. Let $f_0$ be an irreducible divisor of $f$. It suffices to find $g$ such that $f_0^2$ divides $f_0(g(x))$ (which, in turn, divides, $f(g(x))$). Try to choose $g(x)=x+h(x)f_0(x)$. Then $f_0(g(x))=f_0(x+h(x)f_0(x))\equiv f_0(x)+f_0'(x)h(x)f_0(x) \pmod {f_0^2(x)}$, and we need $1+f_0'(x)h(x)$ to be divisible by $f_0$. Since $f_0'$ and $f_0$ are ...


10

The generating function is $$\prod_{i \ge 0}\prod_{j \ge 0} \left(1+z^{2^i 3^j}\right),$$ which, by uniqueness of binary expansion, simplifies to $$\prod_{k \ge 0} \frac{1}{1-z^{3^k}},$$ the generating function of partitions into powers of $3$. See https://oeis.org/A062051


6

Yes. This is normally expressed in terms of the Newton polygon of the polynomial. Specifically, given an arbitrary field $K$ with a valuation, and a polynomial $f(x) = a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0$, the Newton polygon of $f$ is the lower side of the convex hull of the set of points $(i, v(a_i))$. If $K$ is algebraically closed, then the ...


5

I have a strong impression that something like that has been asked before (perhaps, by somebody else) but it is easier to answer again than to find that old thread. What you ask for is patently impossible. Indeed, assume that you have a polynomial $f_n(z)$ that approximates $\sqrt z$ on the interval $[\frac 13,1]$ with precision $e^{-cn}$. Consider the ...


5

This is not an answer to your question, but will point you toward work on the number theoretic properties of such sequences. Iteration of $x^2-x+1$ starting at $a=2$ is called the Sylvester sequence. A theorem about primes that divide the terms in such sequences was proved by Rafe Jones (The density of prime divisors in the arithmetic dynamics of quadratic ...


1

This can be achieved after appropriate changes of coordinates of the source and target. More precisely, there are automorphisms $A, B$ of $\mathbb{C}[x,y]$ such that $A \circ M \circ B$ has the property you want. This follows e.g. from Orevkov's result that if $\tilde M: \mathbb{C}^2 \to \mathbb{C}^2$ is a counterexample to the Jacobian conjecture, then the ...


1

Regarding asymptotics of real zero $\alpha_n$ ($2/r_{2n-1}$ in OP's notation) of generalized Bessel polynomials (with an extra parameter $a$ specialized in the current case to $a=2$), see M.G. de Bruin, E.B. Saff’ and R.S. Varga, On the zeros of generalized Bessel polynomials, Proceedings A, 84 (l), 20, (1981). In theorem $7.3$ they state Obviously, $2/\...


1

The undergrad course I took which included B-splines spent a lot of time first on Bézier curves. You might not necessarily want to spend much time on them, but I think they can motivate a variant on method 1 by defining Bézier curves with de Casteljau's algorithm and B-spline curves with de Boor's algorithm. This assumes that you're at least as interested in ...


1

An answer to this question can be found in the survey by Petter Bränden, https://arxiv.org/abs/1410.6601 Corollary 8.6, where such matrices are described.


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