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)
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discriminant to distinguish parallel line and double line degenerate conic sections
A real affine conic section is the zero locus in $\mathbb{R}^2$ of the quadratic form $$q(x,y)=ax^2+2bxy+cy^2+2dx+2ey+f=0.$$
We may understand this as the $Z=1$ affine patch of the locus in the ...
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Finding the discriminant of a quaternion algebra
Consider the totally real number field $ F=\mathbb{Q}(\zeta_{10}+\zeta_{10}^*) $. Consider the quaternion algebra $ Q=(\frac{-1,-1}{F}) $. How do I compute the discriminant of this algebra?
I gave ...
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convert a 1-dimensional set of points to a 2-dimensional parabola with explicit embedding
I am trying to rephrase to better understand concepts regarding discriminant functions for classification using explicit embedding. I report a very easy diagram found online that from 1-dimension ...
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Smallest possible value of $k$ such that the roots of $x^2-127x+k=0$ are positive integers [closed]
In a triangle, two sides have equal lengths both shorter than the third side. The length of the three sides are all integers and all satisfy the equation $x^2-127x+k=0$, $k$ is a constant. Find the ...
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would the three answers of this integral $\int \frac{1}{a x^2+b x+c} d x$ make a continuous function
so this integral $$
\int \frac{1}{a x^2+b x+c} d x
$$
has three answers depending on the discriminant $$b^2-4 a c$$ whether is positive zero or negative
$b^2-4 a c=0$
$$
\frac{-2}{2 a x+b}
$$
$b^2-4 ...
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Finding $\small{\max\limits_{ab+bc+ca=3}\frac{ab}{2a+b}+\frac{bc}{2b+c}+\frac{ca}{2c+a}.}$
Let $a,b,c \ge 0: ab+bc+ca=3.$ Find maximal value $$P=\frac{ab}{2a+b}+\frac{bc}{2b+c}+\frac{ca}{2c+a}.$$
My tryings lead to wrong inequality.
I tried to use C-S $$\sum_{cyc}\frac{ab}{2a+b}\le \sum_{...
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Mysterious polynomials in the coefficients of $z^4 + bz^3 + cz^2 + dz + e$
I recently started with the polynomial $$P(z) = z^4 + bz^3 + cz^2 + dz + e$$
and calculated an expression closely related to the discriminant of the polar derivative of this polynomial. This ...
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Manipulating multiplicity of polynomial roots based on coefficients
For context, I am trying to solve a ballistics problem in 3D where I want to find the minimum launch velocity of a projectile to be able to hit a moving target. The details can be found here.
I ...
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What is the discriminant of equation, ax^3 - bx + c
Need to find parameters for variables 'a' 'b' 'c' that lead to the graph possessing exactly 2 x axis intercepts and for that I need the discriminant of this equation
Harry
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$\sqrt{d_K}$ lies in $K$?
For a number field $K$, why does the square root of the discriminant $d_K$ lie in the field $K$, i.e., $\sqrt{d_K} \in K$?
I know that $d_K$ is an element of $\mathbb{Q}$, so surely it lies in $K$. In ...
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How to prove this 6-var inequality
Let $a$, $b$, $c$, $x$, $y$, $z$ be real numbers and $ab+bc+ca>0$. Prove that\[\small\left(ab+bc+ca\right)\left(\frac{xy}{\left(a+c\right)\left(b+c\right)}+\frac{yz}{\left(a+b\right)\left(a+c\right)...
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Finding the solutions of $xp^2-2py+4x=0$
I am trying to find the general solution, $p$ and $c$ discriminant and hence the singular solutions (if any) of the following D.E. where $p=\mathrm dy/\mathrm dx$. $$xp^2-2py+4x=0$$
The obvious thing ...
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High school level question of two quadratic equations sharing a common root.
The quadratic equation is
$$
(a^2+b^2)x^2 - 2b(a+c)x + (b^2+c^2) = 0
$$
$a$, $b$, and $c$ are non-zero, real and distinct numbers.
This equation has non-zero real roots $(D \geq 0)$.
One of the roots ...
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Discriminant of a Quaternion Algebra over a local field
I am reading Voight's "Quaternion Algebras" and I have a problem
with Example 29.7.6. which is about the discriminant of a Quaternion Algebra $B$ over a local field $F/K$.
The equation is
$$...
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Discriminant of Orders in Quaternion Algebra "to divide is to contain"
I'm reading Bergeron's book "The Spectrum of Hyperbolic Surfaces". In chapter 8, he talks about orders of quaternion algebras. I am interested in Proposition 8.3 which states
Let $\mathcal{...
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What's so special about the discriminant in the form $b^2 - 4ac$?
The following is the quadratic formula derived by completing the square:
$$x = -\frac{b}{2a} \pm \sqrt{ \frac{b^2}{4a^2} - \frac{c}{a}} \tag{1}$$
This equation is equivalent:
$$x = \frac{-b \pm \sqrt{...
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The discriminant of a certain Appell sequence of polynomials
Consider the following sequence of polynomials
$$p_n(x) = x^n + \binom{n}{1} a x^{n-1}+ \binom{n}{2} a(a+h) x^{n-2} + \cdots + a(a+h) \cdots ( a+ (n-1) h) = \sum_{k=0}^n \binom{n}{k} a^{(k)} x^{n-k} $...
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Discriminant of a polynomial and power sums polynomials
Let $f(x)=x^n+a_1x^{n-1}+\dots+a_n$ be a monic polynomial with coefficients in a field $K$. Its discriminant is defined like this:
$$\Delta(f)=\prod_{i<j}(x_i-x_j)^2$$
Now we define the Newton's ...
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Wikipedia’s solution of extending Fisher LDA
The Wikipedia article on the Kernel Fisher Discriminant Analysis (https://en.wikipedia.org/wiki/Kernel_Fisher_discriminant_analysis) states under the heading “Extending LDA” that $\alpha=N^{-1}\cdot(...
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Relationship between discriminant of polynomials
Just for curiosity, since I haven't found a justification around. Given a polynomial $P(x)=x^{n} − e_{1}x^{n−1} + e_{2}x^{n−2}- \dots + (−1)^{n}e_{n}$ of degree $n$ with discriminant $\Delta_x(P)=f(e_{...
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Is there an easy way to tell whether a cubic or quartic polynomial is factorable over the integers?
For a quadratic, it is easy to tell whether it is factorable. If the discriminant is a perfect square, the quadratic is factorable. Otherwise, the quadratic is not factorable. Is there anything ...
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Question on the discriminant of a irreducible polynomial
Let $\mathbb{K}_5$ be the field with five elements. I don't know to describe the splitting field of $f(x) = x^{3} + x + 1\in\mathbb{K}[5]$ in terms of a root and the discriminant. Clearly, $f(x)$ is ...
user1120269
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Why is $-4a^3c + a^2b^2 + 18abc - 4b^3 - 27c^2$ the discriminant of an elliptic curve?
We have an elliptic curve in its normal form:
$$y^2 = f(x) = x^3 +a x^2 + bx + c,$$
where $a,b,c$ are rational numbers. The discriminant here is said to be
$$-16(4b^3 + 27c^2) \quad \text{ or } \quad ...
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Prove that element is a square (follow up).
I asked the following question and got some awesome answers:
Suppose that $x^5 + ax + b \in \mathbb{F}_p[x]$ is irreducible over $\mathbb{F}_p$. Is it true that $25b^4 + 16a^5$ is a square in $\...
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Theorem of Hecke on the discriminant of a quadratic field extension
Let $k$ be a number field and let $\mathfrak{c}$ be an ideal of $k$ dividing $2\mathcal{O}_k$.
For an ideal $\mathfrak{a}$ of $k$, write $s(\mathfrak{a})$ for the squarefree part of $\mathfrak{a}$, i....
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Proving a discriminant function. Finding boundary of classification.
By Bayes Theorem we have that the posterior probability
P(t|x) for the class t is given by:
P(t|x) = $\frac{P(x|t)P(t)}{P(x|-1)P(-1)+P(x|+1)P(+1)}$
where the priors P(t) satisfy that P(-1)+P(+1)=1. if ...
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If a hidden quadratic has no real roots, does that mean that the equation it represents also has no real roots?
Say you have the equation $y = 9x^4 + 7x^2 + 2$. There are multiple ways of finding the roots of this equation, but one of them is to let $u = x^2$, then re-write the equation as $y = 9u^2 + 7u + 2$, ...
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Why is a field extension separable if and only if the discriminant of the basis of the field extension is nonzero?
Let $L/K$ be a finite dimensional field extension. We define the trace function $T_{L/K}(x)$ of $L$ over $K$ as $T_{L/K}(x)=\textrm{trace}(r_x)$, where $x\in L$ and $r_x$ is matrix given by the ...
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Discriminant of $R^2 \rightarrow R^2$ map
I want to calculate the discriminant set of a function (germ) $f : \mathbb{R}^2 \rightarrow \mathbb{R}^2$ based on the $\mathbb{R} \rightarrow \mathbb{R}$ example of this article. The function is ...
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Sign of fundamental unit in real quadratic number fields with 1 mod 4 discriminant factors
Let $K$ be a real quadratic number field of discriminant $D$ with fundamental unit $\varepsilon$. Further, I want to assume that each positive factor $n$ of $D$ satisfies $n \equiv 1 \pmod 4$. (For ...
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Do real quadratic number fields with prime discriminants have odd class numbers? [closed]
Using the software SageMath I confirmed that there is no real quadratic number field of prime discriminant $D<10^6$ with an even class number. Is this only true ...
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Trace of norm ideal
Let $I$ be a fractional ideal of a real quadratic number field $K$ of discriminant $D$. I thought a little bit about traces of ideals and want to ask if the following is correct. I have a proof for ...
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Can we make the inequality $A-B \leq \frac{A^2}{4}$ strict?
I have an interesting problem:
Given that $$
A=\frac{1}{2022}\left(1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{2022}\right)$$
$$B=\frac{1}{2023}\left(1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{2023}\...
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Extracting the leading coefficient of a discriminant
Let $p(y)$, $q(y)$ be irreducible polynomials in $y$, of the same degree $d$ (over some field $K$). Assume also that the resultant of $p$ and $q$ is $1$ or $-1$ (not sure if it is necessary here).
I'm ...
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Does a discriminant condition on $f(x,y)$ imply that $f$ is weighted homogeneous?
[I cross-posted this question as https://mathoverflow.net/questions/431454. (That version is also slightly updated.)]
Let $f = \sum_{i=m}^n f_iy^i \in \mathbb{C}[x,y]$ be a polynomial (where $f_i \in \...
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Equation has two distinct solutions
We have $\displaystyle{ a,b,c }$ real numbers such that $\displaystyle{ a^2+b^2+c^2>0 }$. Which condition must hold so that the equation $\displaystyle{ ax^2+bx+c=0 }$ has two different solutions?
$...
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Finding Lattice points on a Cubic
I want to study the rational points on a cubic. Eventually I found Nagell's algorithm from http://webs.ucm.es/BUCM/mat/doc8354.pdf, but I cannot immediately apply it because I don't know a rational ...
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Why do we need to include the factor $a_{n}^{2n-2}$ in the discriminant of a polynomial?
Question: Why do we need to include the factor $a_{n}^{2n-2}$ in the discriminant of a polynomial?
Here is the definition of the discriminant ($\Delta$) in terms of the roots $r_1,r_2,...$:
$$ \Delta=...
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Discriminant of $f(x^n)$
I read about this post: Discriminant of $f(x^n)$ for $f$ a quadratic about $\Delta f(x^n)=x^{2n}-bx^n+c$ if $f(x) = x^2-bx+c$ is a quadratic. In particular, $\Delta f(x^n) = n^{2n}c^{n-1}\Delta(f(x))$....
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Prove that the roots of $g(x)$ are also real
Suppose there are two equations $$f(x)=x^2+bx+c=0$$ and $$g(x)=x^2+bx+c(x+a)(2x+b)=0$$
It is given that $f(x)$ has two real roots. Then prove that $g(x)$ also has two real roots when $a\in \mathbb{R}$....
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Find and classify bifurcation points for a cubic ODE
I have an ODE which can be written as
$x' = g(x) = x^3 + px + q$
where $p=-\frac{3c}{A}$ and $q=-\frac{3d}{A}$ (we can assume $c>0,d\neq. 0$) and I am trying to find and classify the bifurcation ...
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Finding $k$ such that the discriminant $k^2 - 12k + 52$ yields a quadratic with only one solution
The given discriminant (not the quadractic) is
$k^2 - 12k +52$
The exact question asks "find value of $k$ when there is only one solution" - that would mean the equation would equal $0$:
$$k^...
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Nature of $\Delta$ in polynomials$?$
Suppose there is a function $f(x)=a_nx^n+a_{n-1}x^{n-1}+a_{n-2}x^{n-2}+\cdots +a_1x+a_0$ and it is given that $a_n>0$ and $f(x)>0\:\: \forall x\in \mathbb{R}$ Then will it be accurate to say ...
user1080568
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Is the theory of $\Delta\le0$, true in cubic functions$?$
If $ax^2+\frac{b}{x}\ge c$ $\forall x>0$ where $a>0 \:\: , b>0$ Show that $27ab^2\ge4c^3$
My work:
Let a function $f(x)$ be $$ax^2+\frac{b}{x}\ge c$$ or we can rewrite $f(x)$ as $$ax^3-cx+b\...
user1080568
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Discriminant in a cubic extension
There is a question: Compute the discriminant $\triangle(1,\alpha,\alpha^2)$, relative to $\mathbb{Q}(\alpha)$, where $\alpha$ is a root of the reducible cubic $x^3+px+q$, $p,q\in\mathbb{Q}$. Is this ...
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Cauchy Schwarz Inequality for Vectors - Quadratic form Clarification
In one of the proofs of Cauchys Inequality that reduces vector inner product to a quadratic form:
$$0 \leq \langle u + \alpha, u + \alpha \rangle = \langle u , u \rangle + 2 \alpha\langle u ,v \...
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Why do we assume the discriminant to be greater than or equal to zero while calculating the range of a function?
Since I don't have enough reputation to comment, I am asking this question again.
I cannot understand why we can assume that the quadratic has real roots and then say $D\ge0$. The answer states that $...
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Least Squares Method Matrix when discriminant = 0
So I have a math question regarding the least squares method.
Given
$$A=\begin{bmatrix}-1&2\\0&0\\1&-2\end{bmatrix}$$
$$b=\begin{bmatrix}1\\-2\\3\end{bmatrix}$$
I want to derive the least-...
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Proof regarding principal factors of the discriminant in $\mathbb{Q}(\sqrt{d})$
So I understand there are (up to $\pm$) exactly two primitive (no rational integer factors) elements $\alpha_1 ,\alpha_2 \in \mathcal{O}_K$ such that the fundamental unit $\varepsilon$ of $K=\mathbb{Q}...
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Discriminant of depressed cubic [duplicate]
If the cubic equation $x^3+px+q$ has roots $\alpha , \beta , \gamma $ then we know that $\alpha + \beta + \gamma =0 $, $\alpha \beta + \alpha \gamma + \beta \gamma =p $ and $\alpha \beta \gamma =-q $.
...
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