Showing papers on "Generic polynomial published in 2021"
TL;DR: A novel, remarkably neat, differential algorithm is introduced, which is suitable to evaluate all the zeros of a generic polynomial of arbitrary degree N.
Abstract: A novel, remarkably neat, differential algorithm is introduced, which is suitable to evaluate all the zeros of a generic polynomial of arbitrary degree N.
3 citations
TL;DR: In this article, the universal factorization algebra of the generic polynomial of degree n into the product of two monic polynomials, one of degree r, and the other of degree c, is described.
Abstract: The goal of this paper is to supply an explicit description of the universal factorization algebra of the generic polynomial of degree n into the product of two monic polynomials, one of degree r, ...
3 citations
TL;DR: A sufficient condition is given for a polynomial f ∈ Q [ x ] to have only trivial multiplicative relations between its roots, which is a generalization of those sufficient conditions proposed in Smyth (1986), Baron et al. (1995) and Dixon (1997) .
2 citations
TL;DR: In this paper, it is known that points, whose fiber under a given polytopes has codimension one, form a finite set of points in a polynomial map.
Abstract: For any two integers $k,n$, $2\leq k\leq n$, let $f:(\mathbb{C}^*)^n\rightarrow\mathbb{C}^k$ be a generic polynomial map with given Newton polytopes. It is known that points, whose fiber under $f$ has codimension one, form a finite set $C_1(f)$ in $\mathbb{C}^k$. For maps $f$ above, we show that $C_1(f)$ is empty if $k\geq 3$, we classify all Newton polytopes contributing to $C_1(f)
eq \emptyset$ for $k=2$, and we compute $|C_1(f)|$.
1 citations
TL;DR: The arboreal Galois group of a polynomial f over a field K encodes the action of Galois on the iterated preimages of a root point x 0 ∈ K, analogous to the l-power torsion of an abelian variety.
TL;DR: In this article, the authors considered quotients of the group algebra of the $3$-string braid group $B_3$ by generic polynomial relations on the elementary braids.
Abstract: We consider quotients of the group algebra of the $3$-string braid group $B_3$ by $p$-th order generic polynomial relations on the elementary braids. In cases $p=2,3,4,5$ these quotient algebras are finite dimensional. We give semisimplicity criteria for these algebras and present explicit formulas for all their irreducible representations.
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TL;DR: In this article, it was shown that the shifted dynatomic polynomials with generalized cyclotomic factors have an interpretation in terms of new multiplicative relations between dynamical units which are uniform in the polynomial $f(x).
Abstract: Given a generic polynomial $f(x)$, the generalized dynatomic polynomial $\Phi_{f,c,d}(x)$ vanishes at precisely those $\alpha$ such that $f^c(\alpha)$ has period exactly $d$ under iteration of $f(x)$. We show that the shifted dynatomic polynomials $\Phi_{f,c,d}(x) - 1$ often have generalized dynatomic factors, and that these factors are in correspondence with certain cyclotomic factors of necklace polynomials. These dynatomic factors of $\Phi_{f,c,d}(x) - 1$ have an interpretation in terms of new multiplicative relations between dynamical units which are uniform in the polynomial $f(x)$.