Root multiplicities and number of nonzero coefficients of a polynomial

Examples on the sharpness of an inequality about multiplicities over hyperfields

In this paper, the author introduces hyperfields and give some facts about the roots and multiplicities of polynomials over hyperfield based on \cite{2} and \cite{3}. Then he tests the sharpness of an inequality in Baker's former work \cite{2} about the behavior of multiplicities under homomorphisms between hyperfields and show that the inequality is not sharp under the natural homomorphisms $\mathbb{C}\rightarrow\mathbb{P}$ and $\mathbb{C}\rightarrow\mathbb{V}$ while it is sharp under the natural homomorphisms $\mathbb{C}\rightarrow\mathbb{K}$, $\mathbb{R}\rightarrow\mathbb{S}$ and $\mathbb{R…

沿多项式的格点可见性密度为一

Density one for lattice point visibility along polynomials with at least two distinct roots

本文证明每个至少有两个不同复根的非零整数多项式沿有理斜率曲线的格点可见性密度为一，通过最大公约数截断将问题化为有限个方程并证明解集密度为零，解决了多项式可见性猜想。

Sharp bounds for the number of roots of univariate fewnomials

Let K be a field and t>=0. Denote by Bm(t,K) the maximum number of non-zero roots in K, counted with multiplicities, of a non-zero polynomial in K[x] with at most t+1 monomial terms. We prove, using an unified approach based on Vandermonde determinants, that Bm(t,L)<=t^2 Bm(t,K) for any local field L with a non-archimedean valuation v such that v(n)=0 for all non-zero integer n and residue field K, and that Bm(t,K)<=(t^2-t+1)(p^f-1) for any finite extension K/Qp with residual class degree f and ramification index e, assuming that p>t+e. For any finite extension K/Qp, for p odd, we also show th…

Roots of Sparse Polynomials over a Finite Field

For a $t$-nomial $f(x) = \sum_{i = 1}^t c_i x^{a_i} \in \mathbb{F}_q[x]$, we show that the number of distinct, nonzero roots of $f$ is bounded above by $2 (q-1)^{1-\varepsilon} C^\varepsilon$, where $\varepsilon = 1/(t-1)$ and $C$ is the size of the largest coset in $\mathbb{F}_q^*$ on which $f$ vanishes completely. Additionally, we describe a number-theoretic parameter depending only on $q$ and the exponents $a_i$ which provides a general and easily-computable upper bound for $C$. We thus obtain a strict improvement over an earlier bound of Canetti et al.\ which is related to the uniformity o…

Multiplicative approximation by the Weil height

Let $K/\mathbb{Q}$ be an algebraic extension of fields, and let $α\not= 0$ be contained in an algebraic closure of $K$. If $α$ can be approximated by roots of numbers in $K^{\times}$ with respect to the Weil height, we prove that some nonzero integer power of $α$ must belong to $K^{\times}$. More generally, let $K_1, K_2, \dots , K_N$, be algebraic extensions of $\mathb{Q}$ such that each pair of extensions includes one which is a (possibly infinite) Galois extension of a common subfield. If $α\not= 0$ can be approximated by a product of roots of numbers from each $K_n$ with respect to the Wei…

多项式计算模 p 平方根的结构与次数

The Structure and Degrees of Polynomials Computing Square Roots $\mod p$

本文研究了模 p 下计算平方根的多项式结构与次数，探讨了在 p≡1 mod 4 时多项式计算平方根的最小次数问题。

Root Repulsion and Faster Solving for Very Sparse Polynomials Over $p$-adic Fields

For any fixed field $K\!\in\!\{\mathbb{Q}_2,\mathbb{Q}_3,\mathbb{Q}_5, \ldots\}$, we prove that all polynomials $f\!\in\!\mathbb{Z}[x]$ with exactly $3$ (resp. $2$) monomial terms, degree $d$, and all coefficients having absolute value at most $H$, can be solved over $K$ within deterministic time $\log^{7+o(1)}(dH)$ (resp. $\log^{2+o(1)}(dH)$) in the classical Turing model: Our underlying algorithm correctly counts the number of roots of $f$ in $K$, and for each such root generates an approximation in $\mathbb{Q}$ with logarithmic height $O(\log^3(dH))$ that converges at a rate of $O\!\left((1…

绝对根分离

Absolute root separation

该研究针对整系数多项式的绝对根分离问题，改进了现有相关下界，并通过低次数实验发现当前界仍偏保守。

Descartes' rule of signs, Newton polygons, and polynomials over hyperfields

We develop a theory of multiplicities of roots for polynomials over hyperfields and use this to provide a unified and conceptual proof of both Descartes' rule of signs and Newton's "polygon rule".

纽结Q多项式的马勒测度与根分布

On the Mahler measure and root distribution of the $Q$-polynomial of links

该研究探讨纽结Q多项式的马勒测度与根分布，证明扭转下Q多项式马勒测度收敛、多数根趋近[-2,2]，对比三类多项式根的差异，给出仅含非零实根的2桥纽结无限族并提出相关猜想。