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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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Relating discriminants of hyperelliptic curves to discriminants of their defining polynomials

Let $C$ be a hyperelliptic curve defined by an equation of the form $$ C: y^2=f(x) $$ where $f$ is a polynomial of prime degree $p\geq3$, over a complete field $K$ of residue characteristic $p$. ...
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Discriminant of quadratic polynomial through the resultant of (f,f')

I would like to calculate the discriminant of $f := ax^2+bx+c$ using the resultant of $(f, f')$. The formula I found for this is $a_n^{-1}(-1)^{\frac{n(n-1)}{2}} \text{res}_x(f(x), f'(x))$. $\text{res}...
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Finding $b$ such that $\dfrac{1}{\left(x - 2\right)^2} + b = -x$ has exactly two solutions, using a non-calculus approach

The question: Let $f(x) = \dfrac{1}{\left(x - 2\right)^2} + b.$ Find $b$ when $f(x) = -x$ has only two solutions. My attempt at a solution (without using any calculus): $$\dfrac{1}{\left(x - 2\right)^...
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Simplifying a quadratic expression under square root

I am trying to simplify the following expression $$\sqrt{R}:=\sqrt{a^2(u_1-u_2)^2+b^2(u_1+u_2)^2-2 ab (u_1^2+u_2^2-2)}$$ I have been staring at it for a while in the hopes of getting rid of the square ...
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Polynomial maximizing its discriminant

Consider a polynomial $p(x) = (x-x_0)(x-x_1)\dots (x-x_{n-1})(x-x_n)$, with $-1 = x_0 < x_1 < \dots < x_{n-1} < x_n = 1$. What values for the roots $x_1, \dots x_{n-1}$ maximize the ...
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Calculating the discriminant of a quintic polynomial

I am trying to calculate the discriminant of the polynomial $f = X^5 + pX^2 + q \in K[X]$ where $K$ is a field. I define the discriminant as $\text{disc } f = \prod_{i < j} (\alpha_i - \alpha_j)^2$ ...
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Galois group of $x^3+x^2-2x-1$ over $\mathbb{Q}$ [duplicate]

Find the Galois group of $p(x)= x^3+x^2-2x-1$ over $\mathbb{Q}$ Note that $p(x)$ is irreducible over $\mathbb{Q}$ by the rational root test. I'm trying to avoid using Cardano/resolvent/discriminant ...
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Determinant of matrix is the discriminant

Let $f = t^n - \sum_{i = 0}^{n-1} a_i t^i \in k[t]$ a polynomial ($k$ a field). I construct a $2n \times 2n$ matrix $A$ in the following way: The upper-left quarter is the $n \times n$-identity ...
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Discriminant of the Kummer extension of p-adics

Let $u\in \mathbb{Z}_p^*$ be a unit in the ring of $p$-adic integers. Assume that $u^{1/p}\not\in \mathbb{Q}_p$, in other words $u$ is not a $p$-power. Define by $K:=\mathbb{Q}_p(u^{1/p})$. What is ...
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Defining discriminant of a polynomial

As per Wikipedia, the discriminant of a polynomial $f(x)=a_nx^n+$(lower degree terms) is defined in terms of roots of $f(x)$ as $$ D(f)=a_n^{2n-2}\prod_{i<j} (\alpha_i-\alpha_j)^2. $$ My question ...
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How do repeated roots of the discriminant correspond to the order of repeated roots in the original polynomial?

Background/context: If $S \subset \mathbb{CP}^1 \times \mathbb{CP}^1$ is a smooth curve with bidegree $(d_1, d_2)$, we know that its genus is $(d_1 - 1)(d_2 - 1)$ for example by the adjunction formula....
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Confusion over defining discriminants in algebraic number theory

I am reviewing some algebraic number theory, and found myself confused about two seemingly distinct notions of discriminant that are used. Let $K$ be a field, and $L$ a finite separable extension. We ...
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Probability that $b^2 - 4ac \geq 0$ where $a,b,c$ are normally distributed (numerical integration)

I would like to determine the probability that a random quadratic polynomial has positive discriminant, where the 3 coefficients $a, b, c$ are normally distributed and independent: That is, given $a,...
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Calculate the discriminant of $X^n+aX^{n-1}+b$, given the discriminant of $X^n+aX+b$

Calculate the discriminant of $X^n+aX^{n-1}+b \in \mathbb{Z}[X]$, knowing that the discriminant of $X^n+aX+b$ is $(-1)^{\frac{n^2+n-2}{2}}(n-1)^{n-1}a^n+(-1)^{\frac{n(n-1)}{2}}n^nb^{n-1}$. All I have ...
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Quintic polynomial with positive discriminant

This is a sequel to that question where I learned why a real quintic polynomial with positive discriminant has either one or five distinct real roots. As a followup, given such a polynomial (i.e. real,...
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Dscriminant for a poynomial of degree higher than 3

I am dealing with a polynomial of degree 5, which I want to prove is positive on a certain interval. I found a statement on Wikipedia that exactly enable me to do this. There is no reference backing ...
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Computing discriminant of an elliptic curve

Let $E$ be the elliptic curve over $\mathbb{Q}$ defined by $y^2+y=x^3-x$. Show that the discriminant $\Delta=37$. Attempt: For an elliptic curve of the form $y^2=x^3+Ax+B$, the discriminant is $4A^3+...
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Why does $\sqrt{3}x^2-4\sqrt{3}=0$ not follow discriminant rules?

From what I understand: $D > 0$ and a perfect square $\Longrightarrow$ Real and Rational Roots $D > 0$ but not a perfect square $\Longrightarrow$ Real and Irrational Roots $D = 0$ $\...
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Discriminant of $\sum_{k=0}^n x^k$

Please help me find discriminant of $\sum_{k=0}^n x^k$. I guess we should find Res($\sum_{k=0}^n x^k, \sum_{k=1}^n kx^{k-1}$) by counting an appropriate Sylvester's determinant, but I can't find a ...
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the degree of the discriminant of a cubic irreducibble polynomial over the rational adjoint a root

Let $f(x) \in \mathbb Q[x]$ and $$f(x)= x^3 + px +q=(x-r_1)(x-r_2)(x-r_3),$$ and $r_1, r_2,r_3\in K$ for the splitting field $K$ of $f(x)$. Let the $D = (r_3 - r_1)^2(r_2- r_1)^2(r_3 - r_2)^2$, which ...
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Prove the sign and zeroes of $Ax^2 + 2Bxy + Cy^2$ (without using the second derivative test)

Let $$P(x,y) = Ax^2 + 2Bxy + Cy^2 \\A \neq 0 \quad x,y \in \mathbb R.$$ Without using the second derivative test, prove that If $AC - B^2 > 0$, then (i) $P$ has no zeroes outside the origin and (...
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Discriminant of $x^n+a$

The polynomial discriminant of $x^2+c$ is $-4c$. The discriminant of $x^3+q$ is $-27q^2.$ Is it true then the discriminant of $x^n+a$ is $-n^na^{n-1}$?
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Why do we need $N$ large in Marcus $3.21$?

Exercise $3.21$ of Marcus’ “Number Fields” (an excellent book) has been giving me trouble on and off for a while now. I think I’ve finally got it but I’m suspicious of my solution since it nowhere ...
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Discriminant of numbers

In a previous post on a $q$-analog of number theory I present the fact that any $q$-number can be written as a unique product of cyclotomic polynomials, similar to the fundamental theorem of ...
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Help to prove $\frac{a^3b+b^3c+c^3a}{a+b+c}+1\ge 2\sqrt{\frac{a^2b^2+b^2c^2+c^2a^2}{a+b+c}}$.

I deal with a problem belonged RMM magazine without success. It's Let $a,b,c>0: abc=1$. Prove that: $$\frac{a^3b+b^3c+c^3a}{a+b+c}+1\ge 2\sqrt{\frac{a^2b^2+b^2c^2+c^2a^2}{a+b+c}}$$ My idea is ...
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$\frac{1}{9-ab}+\frac{1}{9-bc}+\frac{1}{9-ca} \leq \frac{3}{8}.$

(Crux) Let: $a,b,c>0$ such that $a+b+c=3$. Prove that: $$\frac{1}{9-ab}+\frac{1}{9-bc}+\frac{1}{9-ca} \leq \frac{3}{8}.$$ Solution: Put: $x=bc,y=ca,z=ab$. So the inequality can be rewritten as: $$\...
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Is there a relation between the polynomial discriminant and the formula used to solve equations upto degree 4?

For a quadratic polynomial the polynomial discriminant appears in the quadratic formula however, I found this was not the case for cubic and quartic polynomials. The expressions used in the cubic and ...
Poke_Programmer's user avatar
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Do simple roots on most slices imply a polynomial is square free?

Consider a complex polynomial $p(x,y)$ in two variables. I am interested in what it implies for $p$ if for all $y\in\mathbb C$ the one-variable polynomial $p(\cdot,y)$ has or does not have multiple ...
Joonas Ilmavirta's user avatar
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Discriminant divisible by prime iff minimal polynomials reduction mod p has multiple roots

I am attending a first course in algebraic number theory. We have learned the basics of field extensions, integral closures, norm and trace. I am trying to solve the following problem: Let $K=\mathbb{...
janbmull's user avatar
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find the maximum value of $A$ satisfying the following inequality $3x^2+y^2+1\ge A(x^2+xy+x)$

I would appreciate if somebody could help me with the following problem. For any two natural numbers $x$,$y$, find the maximum value of $A$ satisfying the following inequality $$3x^2+y^2+1\ge A(x^2+xy+...
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Find the range of values k can take for $kx^2 + 2x + 1 - k = 0$ to have two real distinct roots

I'm stuck on this question, and I have the answer but I don't know how to get to it. This is what I've done so far. For the quadratic equation $ax^2+bx+c=0$ to have two real roots, the discriminant ...
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How bistable is my system?

Description: given ODE: $\dot{x} = a + bx +cx^2 +dx^3$, I have mutliple combinations of the coefficients $a,b,c,d$ that I want to understand whether they make a bistable or not system. For this ...
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Extremas of rational functions

Our math teacher told us that for rational functions (where the maximum power of x involved is 2) , we assume it equals to some variable $\lambda$ and create a quadratic in x. An example is $ y = \...
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Discriminant of quadratic equation with 6 parameters

The discriminant of a quadratic polynomial is $$\Delta=\left(x_2 \left(x_3+x_5\right)+x_4 \left(x_1+x_5\right)+x_6\left(x_1+x_3\right) \right){}^2-4 \left(x_1 x_3+x_3 x_5+x_1 x_5\right) \left(x_2 x_4+...
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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 ...
tjun kit min's user avatar
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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_{...
Anonymous's user avatar
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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 ...
Iona's user avatar
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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
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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 ...
Yang Awotwi's user avatar
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2 answers
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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)...
youthdoo's user avatar
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2 votes
1 answer
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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 ...
Nothing special's user avatar
4 votes
3 answers
200 views

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 ...
Divyansh Kalra's user avatar
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3 answers
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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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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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