3
votes
Accepted
Is every finite collection of points in $\mathbb{C}^n$ the solution to a compatible algebraic system?
Consider a set of 6 points in $\mathbb{C}^2$ which do not all lie on a common conic -- for example, $\{ (0,0), (1,0), (2,0), (0,1), (1,1), (2,2) \}$. Then $6 = 2 \cdot 3$, but clearly the set cannot ...
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$P(x)$ with rational coefficients, but $P(P(x))$ with integer coefficients
(Not a solution. Too long to be a comment. I pointed out that Srini's solution is in error, so here is my attempt at fixing it.)
Degree 2 case:
$$f(x) = ax^2 + bx + c \\
f(f(x)) = a^3x^4 + 2a^2bx^3 + ...
- 72.2k
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Proof of this derivative definition for polynomials in a finite field polynomial ring? $f(x)=q(x)(x-\alpha)^2+f'(\alpha)(x-\alpha)+\beta$
Ok, first of all, I don't like how they define it because it is kinda confusing.
First note that the set $B_{\alpha}=\{(x-\alpha)^n\mid n\in\mathbb{N}_0\}$ is a basis of $\mathbb{F}_q[x]$ for all $\...
- 208
2
votes
Accepted
Studying the irreducibility of polynomials
For $g$ you write it as a cubic polynomial in $y$:
$$g = x y^3 + (x + 1) y^2 + (x^4 + x^3) y + x^5 \in K[x]\bigl[y\bigr].$$
Since this is of degree $3$ and $K[x]$ is factorial, it suffices to show ...
- 168k
2
votes
Accepted
Proving Simplicity of Zeros of a Polynomial.
Take $f$ to be the polynomial
$$f(z)=1+z+z^2+\cdots+z^n.$$
Its roots are the $n$-th roots of unity different from $1$. For each of those roots, take a small disk $D_k$ around it, such that
the disks $...
- 192
2
votes
Accepted
computing local cohomlogy of a polynomial ideal
Let me address the first question(s). This depends on how you define the local cohomology modules. A very popular reference is the book Local Cohomology by Brodmann and Sharp. If you use their ...
- 749
2
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Necessary and sufficient condition to at least two cubic polynomial roots have absolute value lower than 1
Let's call your polynomial $f(\lambda)$, and let $C$ be the positively oriented unit circle. Assuming none of the roots are on $C$, the number of roots (counted by multiplicity) inside the unit ...
- 458k
2
votes
Accepted
If $L/K$ is an extension, then how to get an irreducible factor of $f(x) \in K[x]$, when an irreducible factor of it is given in $L[x]$?
As stated, this is wrong. Consider $K=\mathbb Q$, $L=\mathbb Q(\sqrt[3]2)$, $f=X^3-2$ and $g=X-\sqrt[3]2$. Then $\operatorname{Aut}_K(L)=\{\operatorname{id}\}$, so the linked definition of the norm ...
- 1,890
1
vote
If $(x-\lambda)^{2n-2}$ divides $\sum_{i,j}p_{i,j}(x)$, does it divide each $p_{i,j}(x)$?
Double indices are unnecessary here. Anyway, here is a counterexample:
$$p_{1,1} =(x-\lambda)^{2n-2} +1, p_{1,2} =(x-\lambda)^{2n-2} -1$$
where $i \in \{1\}$ and $j \in \{1,2\}$.
- 4,824
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vote
Accepted
A homogeneous polynomial divides a polynomial if and if it divides each homogeneous component?
That‘s correct. First suppose that for all i $G|f_i$, then clearly $G|f$. On the other hand if $G|f$ let $f = G * h$ with $h = h_d + .. + h_0$ with homogenous terms $h_i$. Then $f = G*h_d + .. + G*h_0$...
1
vote
Polynomial $f(x)$ has positive coefficients and all roots as real. How many polynomials formed from terms of $f(x)$ also have only real roots?
Some results from this answer are already in Blue's answer.
Following it, we shall call a polynomial real-rooted, if each its root is real.
We assume that the coefficients $a_0,\dots,a_n$ of the real-...
- 95.9k
1
vote
Accepted
$X$, $Y$, $U$, $V$ are independent on $\{0,1,\ldots,n-1\}$ with $X$, $Y$ uniform and $X+Y \sim U+V$
For $n=12$, we'll construct a counterexample . . .
Claim:$\;$There exist independent random variables $X,Y,U,V$, valued on $\{0,...,11\}$, such that
$X,Y$ are uniform.$\\[4pt]$
$U+V\sim X+Y$.$\\[4pt]...
- 59.5k
1
vote
Show a polynomial in two variables is irreducible
To answer your last question first: It's fine to write $f=q(x,y)y+r(x)$, but not the most helpful for the problem. However, it's not possible to write $f(x,y)=q(x)y^2+r(x)$, because the polynomial on ...
- 1,890
1
vote
Accepted
Infinite number of decompositions into sum of four cubes
If there is at least a single integer solution to,
$$a^3+b^3+c^3+d^3 = N$$
such that $(a+b)(c+d)<0$ and $(a+b)(c+d) \neq -\square$, then we can find an infinite more using a Pell equation. We use ...
- 55.7k
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Proof of this derivative definition for polynomials in a finite field polynomial ring? $f(x)=q(x)(x-\alpha)^2+f'(\alpha)(x-\alpha)+\beta$
By the binomial theorems one gets the Horner-Ruffini formula (that they used, not invented) for long division of a polynomial by a linear factor
$$
f(X)=f(\alpha)+g(X,\alpha)(X-\alpha)
$$
$g$ is a ...
- 129k
1
vote
Necessary and sufficient condition to at least two cubic polynomial roots have absolute value lower than 1
In the book titled, "Periodicities in Nonlinear Difference Equations", E. A. Grove and G. Ladas (2005, CRC Press) give many important results for the stability of polynomials that arise in ...
- 2,772
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