# Tagged Questions

129 views

### Solving polynomials of arbitrary degree

Is there any analogue of the method for expressing roots of polynomials of degree 5 with elliptic and η-functions that generalizes to polynomials of degrees n>5? More specifically: does there exist ...
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### Generalization of the Hermite-Bielher-Kakeya Theorem

A stable polynomial is one with zeros in the upper half plane or lower half plane. Interlacing polynomials are polynomials with only real zeros, where between every two zeros of one polynomial lies a ...
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### Estimating decay of certain trigonometric polynomials

For $p=0,1,2,\dots$ and $n=0,1,2,\dots,$, let $f_{n,p}(z)=\sum_{k=0}^n k^p z^k$ be a sequence of polynomials. Restricted to the unit circle, the functions $g_{n,p}(t):=f_{n,p}(e^{it})$ are ...
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### Finding roots of a recursively given polynomial sequence / eigenvalues of an almost Toeplitz matix

I would like to find the roots of the polynomial sequence given by a recurrence relation as follows: $V_0(x) = 1-a^2$ $V_1(x) = 1-a^2 - x$ $V_{k \geq 2}(x) = (1+a^2 - x)V_{k-1}(x) - a^2V_{k-2}(x)$ ...
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### Real-rootedness, interlacing, root-bounds of a sequence of polynomials

Problem: the number $a(n,k)$ is defined by the following recurrence $$a(n,k)=(k+1)(k+2)\, a(n-1, k)+\frac{(k+1)(k+2)(k+3)}{k} \,a(n-1, k-1),$$ with $a(1,1)=1$ and ...
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### Growth of the recurrence coefficients of orthogonal polynomials

Consider the sequence of measures $$d\mu_N(x)=e^{-NV(x)}dx$$ on the real axis, where $V$ is continuous and satisfies the growth assumption $$\lim_{|x|\rightarrow\infty}(V(x)-2\log|x|)=+\infty.$$ ...
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### Markov-type inequalities with arbitrary exponents

By a Markov-type inequality I mean an inequality of the form $$\| p^{(k)} \| \leq \lambda_{k,n} \| p \|,\quad \forall p \in U_n,$$ for some $\lambda_{k,n} > 0$, where $U_n \subset L^\infty[-1,1]$ ...
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### Positive polynomials

How are polynomials called that are positive for all positive, real arguments, e.g., xy + z? How can one determine if a polynomial has this property?
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### evaluation for homogeneous polynomials

Let $p:=\sum_{n=0}^\infty p_n$ be a polynomial given by its terminating decomposition by means of homogeneous polynomials. For fixed $x\in \mathbb{R}^d$ and an none negative integer $n$, can we find ...
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### exponential scaling and polynomials

Consider the following statement $S_a$, with some parameter $a > 1$ : "If $P(X)$ is a real polynomial such that for every $i\in \{0;1;\ldots;N\}$, $|P(a^i)|< 1$ and $|P(a^0)-P(a^1)|>1$, then ...
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### algebraic vis-a-vis analytic functions

Consider functions $\mathbb{R}^n\to\mathbb{R}$. If the set of algebraic functions is the algebraic closure of the rationals, does this mean that limits of polynomials are algebraic? In particular, are ...
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### Positivity of polynomial sequences via generating series

In this question I address the problem of proving the nonnegativity of a numerical sequence $a_0,a_1,a_2,\dots$ via generating series technique. In the notation $A(x)=\sum_{n=0}^\infty a_nx^n\ge0$ ...
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### Is every polynomial a limit of polynomials in quadratic variables?

Let $d>0$ be even. Consider ${\mathbb{R}}[x_1,\ldots, x_n]_d$, i.e. polynomials of degree $d$. Call a homogeneous polynomial $f$ of degree $d$ a polynomial in quadratic variables if it is of the ...
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### Criteria to determine whether a real-coefficient polynomial has real root?

Given a polynomial equation $x^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0=0$, where $n$ is even and all the coefficients $a_i$ are real, what is the best way to determine whether it has a real root or not? I ...
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### Growth of the “cube of square root” function

Hello all, this question is a variant (and probably a more difficult one) of a (promptly answered ) question that I asked here, at Is it true that all the "irrational power" functions are ...
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### Asymptotic series for roots of polynomials

Let $f(z) = z + z^2 + z^3$. Then for large $n$, $f(z) = n$ has a real solution near $n^{1/3}$, which we call $r(n)$. This appears to have an asymptotic series in descending powers of $n^{1/3}$, ...
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### Polynomials and L^p(R)

As someone who mostly does symbolic computation, I've always been puzzled by the fascination mathematicians seem to have with Lp(R) (for p<∞)? To be more precise, there are no non-trivial ...
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### When does a real polynomial have a pair of complex conjugate roots?

Suppose we have a polynomial function $f(z)=a_0+a_1z+a_2z^2+...+z^n$ with each $a_i$ between 0 and 1. Is there a method to determine if $f$ has a pair of complex conjugate roots? There are many ...