# Show that $\sum_{i=0}^{\frac{p-1}{2}} {{\frac{p-1}{2}}\choose {i}}^2 x^{\frac{p-1}{2}-i}$ is separable

I want to show that $\sum_{i=0}^{\frac{p-1}{2}} {{\frac{p-1}{2}}\choose {i}}^2 x^{\frac{p-1}{2}-i}$ is a separable polynomial over this field.Start with $\binom{\frac{p-1}2}i\equiv_p\binom{2i}i4^{-i} \mod p$.

# 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 $A = \{(x,y) \in \mathbb R^{n+1} :\; p(x,y) = 0\}$.

# A criterion for real-rooted polynomials with nonnegative coefficients

Let $P \in \mathbb{R}[X]$, with $\deg P = n$.Is it true that

$P$ has only real roots $\quad \Longleftrightarrow \quad P\cdot P'' + (\frac{1}{n}-1)P'^2 \leq 0$?

# 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
$$\dot{x}= \frac{\partial H}{\partial y}(x,y) + H(x,y) Q(x,y)$$
$$\dot{y}= - \frac{\partial H}{\partial x}(x,y) - H(x,y) P(x,y). # Fourier coefficients of positive polynomials Let p(x) \geq 0 be a positive polynomial on the hypersphere (x \in S^{n-1}) satisfying \int_{S^{n-1}} p(x) = 1.Are there any bounds known on the components p_j(x) using that p(x) \geq 0 for all x? # Generating primes with floor of a polynomial [p(n)] Is there a polynomial p(x) with real coefficients and degree at least one such that [p(n)] is prime for every natural number n?If yes, what is such a polynomial p(x) and if no, how to prove? # Sum of Squares and Harmonic Functions Let a_0, a_1, \dots a_k be non-negative reals.Any homogeneous polynomial p of degree 2k in \mathbb{R}^{d} can be decomposed as$$
p(x)=\sum_{i=0}^{k}c_i|x|^{2(k-i)}p_{2i}(x)
$$for some c_i, where each p_{2i} is a spherical harmonic of degree 2i. # Consecutive square values of cubic polynomials Let P(x) be a cubic polynomial with integer coefficients.Does there exist a constant c such that at least one of the following values P(0),P(1),. # Ulam stability of homogeneous polynomials Let P be a homogenous polynomial with real coefficients in several variable(at least three variable) Is the following statement true: For every \epsilon there is a \delta such that for every x with |P(x)|< \delta we have d(x,Z)<\epsilon.Here Z=P^{-1}(\{0\}) is the set of roots of P, and d is the standard. # Questions about expansion of f(x)=\sum_{i=1}^{\infty} a_i x^i Question 1: are there$$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 $$f(x)=p(x)+\sum_{i=1}^{n}r_i(x),n <\infty? # Algorithm for representing a polynomial as a composition of lower degree polynomials Let p(x) be a polynomial of degree e^n with coefficients in \mathbb Z_q such that there exists a progression of polynomials (the "composition") p_i(x) = a_i(p_{i-1}(x))^e+b_i, a_i, b_i \ne 0 with p_0(x) = x and p(x) = p_n(x) Given q and p(x), how do I find any such composition?As in my comment, I will assume that p(x) and all p_i(x) are monic. # Every positive polynomial with rational coefficients is above a completely Q-factorized nonnegative polynomial ? Let P be a unitary polynomial with rational coefficients in one variable x, such that P(x) \gt 0 for all x \in \mathbb R.I can show that the answer is YES when d=1 or d=2. # Cyclotomic polynomials coprime to a fixed polynomial Let f \in \mathbb{Z}[x] be monic, irreducible and hyperbolic (no roots of absolute value 1), and such that f(0)= \pm 1.Denoting as c_{p}(x) the cyclotomic polynomial$$c_{p}(x)=1+x+\cdots +x^{p-1},
my question is: how can be characterized the (certainly finite?

# nth-powers and degree n polynomials with coefficients in field extensions

Hi,

Suppose that $E/F$ is a Galois extension.If $P(X)\in E[X]$ is a (EDIT: monic) polynomial of
degree $n > 0$, such that $P(X)^n\in F[X]$, does it follow that $P(X)\in F[X]$?