Interpolation

Find the polynomial of least degree from the given table of its values:

Prove that a polynomial of degree $<n$, which takes integer values at $n$ consecutive integer values of the variable, takes integer values at all integer values of the variable.

Is it true that such a polynomial has integer coefficients?

Prove that every function $f: F \rightarrow F$ on a finite field $F$ of $q$ elements is uniquely represented in the form of a polynomial of degree $<q$.

Prove that the polynomial of degree $<n$ that takes the values $y_{1}, \ldots , y_{n}$ at the points $x_{1}, \ldots , x_{n}$ equals

where $g(x)=\left(x-x_{1}\right) \ldots \left(x-x_{n}\right)$.

A polynomial $f(x)$ of degree at most $n-1$ takes the values $y_{1}, \ldots , y_{n}$ at the roots of degree $n$ of 1. Find $f(0)$.

Prove that the points $x_{1}, \ldots , x_{n} \in \mathbb {C}$ are the vertices of a regular $n$-gon centered at the point $x_{0}$ if and only if for every polynomial $f(x)$ of degree $<n$ the equality

holds.

Let all the roots $x_{1}, \ldots , x_{n}$ of the polynomial $f(x)$ be distinct.

Find a polynomial of degree $2 n$ that leaves the remainder $-1$ upon division by $x(x-2) \ldots (x-2 n)$.

Construct over $\mathbf{Z}_{p}$ a polynomial $f(x)$ of least degree with the condition $f(k)=k^{-1}$ for $k=1,2, \ldots , p-1$.

Construct over $\mathbf{Z}_{7}$ a polynomial $f(x)$ of least degree with the condition $f(0)=1, f(1)=0$ and $f(k)=k$ for $k=2,3,4,5,6$.

Let $\mathbb {F}_{q}$ be a field of $q>2$ elements, $c$ a generator of the cyclic group $\mathbb {F}_{q}^{*}$. Prove that the group of permutations $S_{q}$, realized on $\mathbb {F}_{q}$, is generated by the maps $f(x)=x+1, h(x)=c x$, $g(x)=x^{q-2}$.

In the setting of 30.11, prove that the alternating group $A_{q}$ is generated by the polynomials $c^{2} x, x+1,\left(x^{q-2}+1\right)^{q-2}$.

Let $k_{0}, \ldots , k_{n}$ be natural numbers and $x_{i}, b_{i j}$ elements of a field $F$ of characteristic zero, where $i=0, \ldots , n, j=0, \ldots , k_{i}-1$. Assume that the elements $x_{0}, \ldots , x_{n}$ are distinct. Prove that there exists a unique polynomial $f(x) \in F[x]$ of degree at most $k_{0}+\ldots +k_{n}-1$ such that $f^{j}\left(x_{i}\right)=b_{i j}$ for all $i, j$.

In the setting of problem 30.13, set

where $G_{i}(x)=\prod_{j \neq i}\left(x-x_{j}\right)^{k_{j}}$. Prove that $f(x)$ is a polynomial of degree at most $k_{0}+\ldots +k_{n}-1$ and $f^{(j)}\left(x_{i}\right)=b_{i j}$ for all $i, j$.