Tagged Questions
2
votes
1answer
32 views
$p(x)$ be a polynomial over $\mathbb{Z}$. If $P(a)=P(b)=P(c)=-1$ with integers $a,b,c$.Then $P(x)$ has no integral roots
Let $\mathbb{P}(x)$ be a polynomial over $\mathbb{Z}$.
If $\mathbb{P}(a)$=$\mathbb{P}(b)$=$\mathbb{P}(c)$=$-1$ with integers $a,b,c.$
Then $\mathbb{P}(x)$ has no integral roots
2
votes
0answers
82 views
general biquadratic equation
I'm not sure if I understand what the following question is asking:
Show that the solution of the general biquadratic equation
$x^{4}+ax^{3}+bx^{2}+cx+d=0$ can be obtained directly, that is,
...
3
votes
3answers
85 views
Ring of invariants of Klein Four group
Assume $F$ is a field and assume $f\in F[x_1,\ldots,x_4]$ is a polynomial that is invariant under the Klein Four group $V_4$. How can I show that this polynomial can then be rewritten as a polynomial ...
2
votes
1answer
65 views
Symmetric polynomial optimization
Recently I asked a stupid question here (there’s no harm in that, even Fields medalist Terence Tao advises to ask dumb questions once in a while).
Here is a variant question that may be more ...
1
vote
2answers
87 views
Three inequalities with sums of fractions over two positive integers
In a proof, I arrive at three inequalities for all $p,q \geqslant 0$:
\begin{align}
\frac{p+1}{q+1} + \frac{q+1}{p+1} &\geqslant 1 +
\frac{p}{2q+1} + \frac{q}{2p+1} + \frac{1}{p+q+1};\cr
...
0
votes
0answers
62 views
Showing that an alternating polynomial is the product of some symmetric polynomial and the Vandermonde polynomial
For simplicity, consider polynomials of two variables. Let $f(x,y)$ be an arbitrary alternating polynomial. I want to show that $f(x, y)$ is the product of some symmetric polynomial and the ...
6
votes
1answer
378 views
Number of distinct $f(x_1,x_2,x_3,\ldots,x_n)$ under permutation of the input
$\alpha _n ^n-1=0$
$\alpha _n=e^{2 \pi i/n}$
$$f(x_1,x_2,x_3,\ldots,x_n)=(x_1+\alpha _n x_2+ \alpha _n ^2 x_3+\cdots+\alpha _n ^{n-1} x_n)^n$$
I have read in Jim Brown's paper on page 5 that ...
5
votes
4answers
461 views
Expressing a symmetric polynomial in terms of elementary symmetric polynomials using computer?
Are there any computer algebra systems with the functionality to allow me to enter in an explicit symmetric polynomial and have it return that polynomial in terms of the elementary symmetric ...
9
votes
1answer
902 views
Using Vieta's theorem for cubic equations to derive the cubic discriminant
Background:
Vieta's Theorem for cubic equations says that if a cubic equation $x^3 + px^2 + qx + r = 0$ has three different roots $x_1, x_2, x_3$, then
$$\begin{eqnarray*}
-p &=& x_1 + x_2 ...
3
votes
3answers
147 views
Do these special power functions generate all homogeneous symmetric polynomials?
Over rational numbers, the set of all power functions up to a certain degree generate all symmetric polynomials in that degree.
My question is as follows. To be succinct, let's say we have four ...
4
votes
2answers
429 views
Sum of cubed roots
I need to calculate the sums
$$x_1^3 + x_2^3 + x_3^3$$
and
$$x_1^4 + x_2^4 + x_3^4$$
where $x_1, x_2, x_3$ are the roots of
$$x^3+2x^2+3x+4=0$$
using Viete's formulas.
I know that ...
6
votes
3answers
387 views
Three-variable system of simultaneous equations
$x + y + z = 4$
$x^2 + y^2 + z^2 = 4$
$x^3 + y^3 + z^3 = 4$
Any ideas on how to solve for $(x,y,z)$ satisfying the three simultaneous equations, provided there can be both real and complex ...
6
votes
2answers
185 views
Number of terms in a monomial symmetric polynomial
Is there a closed form expression for the number of terms in a monomial symmetric polynomial in a given number of variables for a particular partition of exponents, in terms of which/how many ...
5
votes
2answers
623 views
symmetric polynomials and the Newton identities
I want to write
$P(x,y,z)=yx^{3}+zx^{3}+xy^{3}+zy^{3}+xz^{3}+yz^{3}$
in terms of elementary symmetric polynomials, but I'm getting stuck at the first step. I know I should follow the proof of the ...
14
votes
2answers
1k views
Why does the discriminant of a cubic polynomial being less than 0 indicate complex roots?
The discriminant $\Delta = 18abcd - 4b^3d + b^2 c^2 - 4ac^3 - 27a^2d^2$ of the cubic polynomial $ax^3 + bx^2 + cx+ d$ indicates not only if there are repeated roots when $\Delta$ vanishes, but also ...
