Results for query "Suppose $p(x)$ is a monic cubic polynomial with real coefficients such that $p(3-2i)=0$ and $p(0)=-52$. Determine $p(x)$ (in expanded form)."

Number of real roots of a polynomial

Let $P\in \mathbb{R}[x]$ be a polynomial such that $(P, P') = 1$.Suppose that we want to calculate the number of real roots of $P$ in the interval $[a, b]$ (to simplify, let us assume that $P(a), P(b) \ne 0$).

Zero measurability of zero-sets of polynomials

Here is a sketch of an argument I have: Let $p(x,y) \equiv p(x,y_1,\dots,y_n)$ be a polynomial of degree $d$ (with real coefficients say) in $n+1$ variables.Let $A = \{(x,y) \in \mathbb R^{n+1} :\; p(x,y) = 0\}$.

order of reality

Suppose I have a monic polynomial $p(x) \in \mathbb{Z}[x],$ of degree $d,$ and $\alpha$ a nonzero root of $p(x).$ The question is: assuming that $\alpha$ has some real power, what is the maximum order of $\alpha$ in $\mathbb{C}^\times/\mathbb{R}^\times?

Solving cubic equations in characteristic 2

Consider the cubic polynomial
f = x^3+px+q,
where $p,q$ are elements of a fixed algebraic closure $\overline{\mathbb{F}}_2$ of $\mathbb{F}_2$.Is there an elegant criterion for deciding whether $f$ has $0$, $1$ or $3$ roots in the field $\mathbb{F}_2(p,q)$?

Transcendency of certain integrals

Let $p$ be an even monic polynomial with rational coefficients of degree at least $4$.Can the integral
$$\int_{-\infty}^\infty e^{-p(x)}dx$$
be an algebraic number?

Two limit cycles which lie on the same leaf

Take an non-singular algebraic curve $H(x,y)=0$ given by a polynomial $H(x,y)$ with real coefficients that has at least two ovals in the affine real plane, and generate the one-form $dH + H \omega$, where $\omega = P(x,y)dx + Q(x,y)dy$ is a real polynomial one-form.The foliation generated by the kernel of the form $dH + H \omega = 0$ gives you the orbits of the vector field
$$\dot{x}= \frac{\partial H}{\partial y}(x,y) + H(x,y) Q(x,y)$$
$$\dot{y}= - \frac{\partial H}{\partial x}(x,y) - H(x,y) P(x,y).

Fourier coefficients of positive polynomials

Let $p(x) \geq 0$ be a positive polynomial on the hypersphere ($x \in S^{n-1}$) satisfying $\int_{S^{n-1}} p(x) = 1$.Are there any bounds known on the components $p_j(x)$ using that $p(x) \geq 0$ for all $x$?

Sum of Squares and Harmonic Functions

Let $a_0, a_1, \dots a_k$ be non-negative reals.Any homogeneous polynomial $p$ of degree $2k$ in $\mathbb{R}^{d}$ can be decomposed as
for some $c_i$, where each $p_{2i}$ is a spherical harmonic of degree $2i$.

Ulam stability of homogeneous polynomials

Let $P$ be a homogenous polynomial with real coefficients in several variable(at least three variable)
Is the following statement true:

For every $\epsilon$ there is a $\delta$ such that for every x with $|P(x)|< \delta$ we have $d(x,Z)<\epsilon$.Here $Z=P^{-1}(\{0\})$ is the set of roots of $P$, and $d$ is the standard.

Questions about expansion of $f(x)=\sum_{i=1}^{\infty} a_i x^i$

Question 1: are there $$f(x)=p(x)+\sum_{i=1}^{\infty}r_i(x), $$
or $$f(x)=p(x)+\sum_{i=1}^{n}r_i(x),$$
where $p(x)$ is a polynomial with all coefficients which are natural numbers, and $r_i(x)$ is a quotient of polynomials with at least one pole(which means denominator is polynomial with at least one zero point),$r_i(x)$ can be expanded as $$r_i(x) =\sum_{j=1}^{\infty} b_{ij} x^j, \text{ }b_{ij} \in N\text{ or } b_{ij} = 0?Question 3: Under what condition $$f(x)=p(x)+\sum_{i=1}^{n}r_i(x),n <\infty?

Algorithm for representing a polynomial as a composition of lower degree polynomials

Let $p(x)$ be a polynomial of degree $e^n$ with coefficients in $\mathbb Z_q$ such that there exists a progression of polynomials (the "composition")

$p_i(x) = a_i(p_{i-1}(x))^e+b_i$, $a_i, b_i \ne 0$ with
$p_0(x) = x$ and
$p(x) = p_n(x)$

Given $q$ and $p(x)$, how do I find any such composition?As in my comment, I will assume that $p(x)$ and all $p_i(x)$ are monic.

Cyclotomic polynomials coprime to a fixed polynomial

Let $f \in \mathbb{Z}[x]$ be monic, irreducible and hyperbolic (no roots of absolute value $1$), and such that $f(0)= \pm 1$.Denoting as $c_{p}(x)$ the cyclotomic polynomial $$c_{p}(x)=1+x+\cdots +x^{p-1},$$
my question is: how can be characterized the (certainly finite?