# Tagged Questions

This tag is used for both basic and advanced questions on polynomials in any number of variables. Including, but not limited to: solving for roots, factoring, checking for irreducibility. This tag is rarely used as the only tag for a question.

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### Derivation of the discriminant of a cubic polynomial by algebraic manipulation.

The problem was asked before: Using Vieta's theorem for cubic equations to derive the cubic discriminant . I tried to solve it by purely algebraic manipulation but was faced with an explosion of ...
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### Monic polynomial

Recently I've learned that when a given polynomial is a monic polynomial, then this polynomial root has to be a rational root. As far as I know, to figure out if a given polynomial is a monic ...
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### What can we learn from prime generating polynomials?

Here's a simple polynomial that generates quite a few primes (not necessarily consecutive). $p(n) = n^2 + 23n + 23$ with $n=0,1,2...$ What can such polynomials tell us about primes? Thanks. ...
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### Relation between roots an coefficients in a generic equation: $a_0+a_1\cdot x+\cdots+a_n\cdot x^n$

In a generic equation $$a_0\cdot x+a_1\cdot x^2+ a_3\cdot x^3+\cdots+a_n\cdot x^n$$ there are some relations between roots ($x_1, x_2,\ldots,x_n$) and coefficients ($a_0, a_1,\ldots,a_n$). How can i ...
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### Condition Butterworth polynomial

My course states that a polynomial is a Butterworth polynomial when it satisfies the following condition: $|B(j\Omega)|=\sqrt {1+{\Omega}^{2\,n}}=\sqrt {1+{(\omega/\omega_p)}^{2\,n}}$ I'm really ...
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### Is there closed form solution for this infinite polynomial or high-order polymonial?

The equation is as follows \begin{align} \sum_{N=1}^{\infty}P(N)x^N=Z, \end{align} where $P(N)$'s are real number satisfying $0\leq P(N)\leq 1$. Another equation is \begin{align} \sum_{N=1}^{\bar N}...
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### why there are not polynomials $p,q$ such that $\sqrt{x^2-4}=\frac{p(x)}{q(x)}$

show that there are not polynomials $p,q$ such that $$\sqrt{x^2-4}=\dfrac{p(x)}{q(x)}$$ there a book say it is clear,because if such polynomials existed,then each zero of$x^2-4$ should have even ...
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### An ideal in a ring of polynomials and a field extension.

Let $K\subseteq L$ be fields and $I$ an ideal of $K[x_1,...,x_n]$. I want to show that $IL[x_1,...,x_n]\cap K[x_1,...,x_n] =I$. The inclusion $I \subseteq IL[x_1,...,x_n]\cap K[x_1,...,x_n]$ is clear,...
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### Inverse pairing function with polynomial constituents

Many bijective pairing functions $f:\mathbb N \times \mathbb N \rightarrow \mathbb N$ exists, including polynomial ones such as the Cantor pairing function $$f(n,m) = \frac{1}{2}(n + m)(n + m + 1)+m$$...
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### $x^3-3x^2+(a^2+2)x-a^2$ has 3 roots $x_1,x_2,x_3$ such that $\sin \tfrac{2\pi x_1}{3}+\sin \tfrac{2\pi x_3}{3}=2\sin \tfrac{2\pi x_2}{3}$. Find $a$.

$x^3-3x^2+(a^2+2)x-a^2$ has 3 roots $x_1,x_2,x_3$ such that $\sin \dfrac{2\pi x_1}{3}+\sin \dfrac{2\pi x_3}{3}=2\sin \dfrac{2\pi x_2}{3}$. Find $a$ (Bulgari 1998)
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### How can I use Fundamental Theorem of Symmetric Polynomials to factor polynomials?

How can I use The fundamental theorem of symmetric polynomials (or its proof) to factor symmetric polynomials? The link I've given to the theorem uses elaborate wordings using 'rings', 'isomorphic'...
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### Existence of polynomials $g$, $h$, with non-negative coefficients, such that $f(x)= \frac{g(x)}{h(x)}$ [closed]

Suppose $a$ and $b$ are real numbers such that the quadratic polynomial $f(x) = x^2 + ax + b$has no non-negative real roots. Prove that ther exist two polynomials g,h, whose coefficients are non-...
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### Shamir's secret sharing interpolation problem

I try to understand this protocol - Shamir's secret sharing - threshold scheme. I got my data and I made interpolation basing on examples published on Wikipedia. You can see them below (sorry, I am ...
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### Prove the extension to be a Galois Extension

Let $p$ be a prime number. $K$=$\mathbb C(x,y)$ and $F=\mathbb C(x^p,y^p)$.Then, Prove that $K/F$ is a Galois Extension. Trial: Since this $\mathbb C$ is a field of charactersitic $0$,it would be ...
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### On the leading coefficient of a polynomial which takes integer values at every integer argument

If $f(x)$ is a polynomial with complex coefficients of degree $k$ with leading coefficient $a_k$ such that $f(n) \in \mathbb Z, \forall n \in \mathbb Z$, then is it true that $|a_k| \ge \dfrac 1{k!}$ ...
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### relations between a set of polynomials

I have a set of polynomials. Is there a computer algebra program that gives all the algebraic relations between them ? I will prefer singular if it has this component.
If I draw a graph of the quadratic $x^2-9=0$, I have a parabola with roots $x=3$ and $x=-3$ and a vertex of $(0,-9)$ with the parabola opening upwards as $a$ is positive in the standard quadratic form....
### Show that $\mathbb{F}[x^2,y^2,xy]$ is not polynomial
$\mathbb{F}[x^2,y^2,xy]$ is the polynomials in two variables whose terms all have even degrees. Of course, this generating set $x^2,y^2,xy$ is not algebraically independent, but I need to show that no ...