Questions tagged [discriminant]
Discriminant of a polynomial $\;P\left(x\right) = a_{0} + a_{1}x + a_{2}x^{2} + \dots + a_{n}x^{n} \neq 0\,$ is defined as \begin{align} \Delta &= a_{n}^{2n-2}\prod_{ i < j } \big( r_i - r_j \big)^{2} = \left(-1\right)^{n\left(n-1\right)/2} a_{n}^{2n-2}\prod_{ i \neq j } \big( r_i - r_j \big) \end{align} where $\,r_1,\dots,r_n\,$ are roots of $P\left(x\right)$ (counting multiplicity)
467 questions
8 votes
1 answer
240 views
Not all quadratic extensions over $\mathbb{Q}$ are contained in the compositum of all the splitting fields of irreducible cubics in $\mathbb{Q}[X]$
Question: Let $F$ be the composite of all the splitting fields of irreducible cubics over $\mathbb{Q}$. Prove that $F$ does not contain all quadratic extensions of $\mathbb{Q}$. (This is exercise 16 ...
- 813
1 vote
3 answers
212 views
If $a,b,c>0$ such that $(a+b+c)^3=125abc$, then show that $ \dfrac{a}{\sqrt{bc}}+\dfrac{b}{\sqrt{ca}}+\dfrac{c}{\sqrt{ab}} \le \dfrac{16+\sqrt{2}}{2}$
Problem. Let $a,b,c$ be positive real numbers such that $$(a+b+c)^3=125abc$$ Prove that : $$ \dfrac{a}{\sqrt{bc}}+\dfrac{b}{\sqrt{ca}}+\dfrac{c}{\sqrt{ab}} \le \dfrac{16+\sqrt{2}}{2}$$ This problem ...
- 111
1 vote
1 answer
181 views
An upper and lower bound of $abc+bcd+cda+dab$
Problem. Let $a,\,b,\,c,\,d$ be real numbers. Denote $$u = \frac{a+b+c+d}{4}, \quad v = \frac{ab+ac+ad+bc+bd+cd}{6}, \quad w = \frac{abc+bcd+cda+dab}{4}.$$ Prove that $$3uv-2u^{3} - 2\left(u^{2}-v\...
- 6,289
2 votes
1 answer
160 views
Meaning of the discriminant of a general Jacobian
Let's say I have two functions: $y(a,b)$ and $z(a,b)$, and I use the following Jacobian: $J=\left(\begin{array}{cc}\frac{\partial y}{\partial a} & \frac{\partial y}{\partial b} \\ { \frac{\partial ...
- 299
0 votes
3 answers
239 views
An upper and lower bound of $abcd$
Problem. Given $a,\,b,\,c,\,d$ be non-negative real numbers. Denote $$u = \frac{a+b+c+d}{4}, \quad v^2 = \frac{ab+ac+ad+bc+bd+cd}{6}.$$ Prove that $$12u^2v^2-8u^4-3v^4-8u(u^2-v^2)\sqrt{u^2-v^2} \...
- 6,289
1 vote
2 answers
121 views
Prove $\frac{1}{(2a+1)(2b+1)}+\frac{1}{(2b+1)(2c+1)}+\frac{1}{(2c+1)(2a+1)} \geqslant \frac{3}{3+2(ab+bc+ca)}.$
Problem. Let $a,b,c$ are positive real numbers. Prove that $$\frac{1}{(2a+1)(2b+1)}+\frac{1}{(2b+1)(2c+1)}+\frac{1}{(2c+1)(2a+1)} \geqslant \frac{3}{3+2(ab+bc+ca)}.$$ The inequality is equivalent to $$...
user1656827
3 votes
2 answers
333 views
A general USA MO 2003
From a famous inequality Problem 1. (USA MO 2003) Let $ a$, $ b$, $ c$ be positive real numbers. Prove that $$\dfrac{(2a + b + c)^2}{2a^2 + (b + c)^2} + \dfrac{(2b + c + a)^2}{2b^2 + (c + a)^2} + \...
- 6,289
0 votes
2 answers
76 views
Third degree polynomial with small parameter
This is a follow-up to that question, so I will refer to it for the motivation. In continuation, I now have this polynomial obtained by inserting $x=\phi_2+\sqrt{\varepsilon}\cdot y$ in the polynomial ...
- 1,395
3 votes
3 answers
294 views
Singular expansion of a root of a polynomial
I find the following issue when dealing with a problem concerning a PDE. Consider the polynomial equation for $x$ given by $$ (1-x)(2b+x^2)=2\phi_2(1-\phi_2)^2,\;\;\frac{1}{3}<\phi_2<{1}. $$ At ...
- 1,395
3 votes
1 answer
209 views
Theorem 3.9 from Janusz - Algebraic Number Fields (Edition 2)
Unramified Extensions The unramified extensions of a completion of a number field at a nonarchimedean prime are easily described and have a number of very special properties. We give just a few ...
- 813
1 vote
4 answers
247 views
Find largest $k$ so that $\left(\sum \frac{a^{2}+bc}{b-c}\right)^{2}\ge k\cdot \left(ab+bc+ca\right)$ is true
I'm looking for some ideas to solve the following inequality. Problem. For any pairwise distinct real numbers $a,b,c$ then find the maximal constant $k$ such that$$\left(\frac{a^{2}+bc}{b-c}+\frac{b^...
- 957
6 votes
2 answers
476 views
Ideas and methods to compute absolute discriminant of fields.
Just for curiosity and fun, I was trying to calculate the discriminant of some unusual fields! Let $p=-3\cdot 11\cdot 29\cdot 31 = -29667$ and $q=2\cdot 5\cdot 11\cdot 19\cdot 29\cdot 31 = 1878910$. ...
- 1,096
3 votes
6 answers
539 views
Prove that $a^2+b^2+c^2+5abc\geq8$
Let $a$, $b$ and $c$ be non-negative numbers such that $$(a+b+c-2)^2+8\leq3(ab+ac+bc).$$ Prove that $$a^2+b^2+c^2+5abc\geq8.$$ This inequality was posted here. My attempts. Let $a+b+c=3u$, $ab+ac+bc=...
1 vote
2 answers
402 views
Prove $\sum\limits_{\mathrm{cyc}} (\frac{c^{2}+a^{2}}{c-a})^{2}\ge \left(4+2\sqrt{6}\right)\left(ab+bc+ca\right)$
Let $a,b,c$ be distinct real numbers. Prove that$$\left(\frac{a^{2}+b^{2}}{a-b}\right)^{2}+\left(\frac{b^{2}+c^{2}}{b-c}\right)^{2}+\left(\frac{c^{2}+a^{2}}{c-a}\right)^{2}\ge \left(4+2\sqrt{6}\right)\...
user1630296
0 votes
0 answers
35 views
Analog to discriminant for ratio of polynomials and fractional powers of polynomials
Is there a quantity/relationship that is analogous to the polynomial discriminant that would be useful in the following context? I am trying to classify the possible solutions to the following ...
