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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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Level Lowering obstructions

For the generalized Fermat equation: $a^p + b^q = c^r$ with $p,q,r\ge3$ and $\gcd(a,b,c)=1$ one can construct the Frey Curve: $y^2=x(x-a^p)(x+b^q)$ which is semi-stable for those primes $m$ ...
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Inequality with square roots where solution found with discriminant is not valid

I have : $3 + \sqrt{x-1} > \sqrt{2x}$ when doing basic algebraic operations I will find that : $ 0 > x^{2} -52x + 100 $ I will use $b^2-4ac$ formula to eventually find out that there are ...
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If $P$ is a real quadratic polynomial, then $|P(z)| \geq \sqrt{\Delta} |Im(z)|$

I was reading an article which contains a statement about the distance in the hyperbolic plane: if $\gamma \in SL_2(\mathbb R)$ is hyperbolic then $\inf_{z \in H} d(\gamma z, z) = arcosh(Tr(\gamma)^2/...
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Find maximum value of $\frac xy$

If $x^2-30x+y^2-40y+576=0$, find the maximum value of $\dfrac xy$. First I completed the squares and got $(x-15)^2+(y-20)^2=7^2$, which is the equation of a circle. I think I need to use some ...
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Show $\Delta(\theta)\mathbb{Z} = \Delta(R/\mathbb{Z}) \implies R = \mathbb{Z}[\theta]$.

Let $\Delta$ denote the discriminant and $\Delta(R/\mathbb{Z})$ denote the discriminant ideal of $R$ over $\mathbb{Z}$. I'm having trouble proving the following: Suppose $[\mathbb{Q}(\theta) : \...
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Discriminant when graph lies above or below the x axis.

Suppose a quadratic equation has been given where the a value (ax^2 + bx + c) is a positive and it has been said that the graph of the equation lies above the x-axis- what is the discriminant? For ...
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Factorisation of fundamental discriminants

$D$ is a fundamental discriminant if $D\equiv 1\pmod 4$ and $D$ is square-free or $D=4m,$ where $m\equiv 2,3 \pmod 4$ and $m$ is square-free. On wikipedia I found the following characterisation: ...
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What happens to the group structure of an elliptic curve over a field when the discriminant = 0?

Working on a question for a number theory class. So, basically, it asks us what happens to the group structure of an elliptic curve over a field if the discriminant is equal to zero? So, basically,...
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Discriminant Integral closure of ring

I have trouble to understand how to generalize the definition of the discriminant $Disc(B/A)$ where $A$ is the ring of integer in some number field $K$ and $B$ the integral closure in some finite ...
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Determining the nature of quadratic equation

If $a,b,c$ are real numbers and $a+b+c =0$ then how to prove that the equation $$4ax^2+3bx+2c$$ has two real roots. I just know that for real roots the quadratic equation should have its Discriminant ...
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Primes representable by equivalent binary forms

So the question I have is this , if a prime p is representable by two primitive positive definitive binary quadratic forms f(x,y) and g(x,y) of the same discriminant, are they necessarily equivalent? ...
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Finding the value of k that satisfies the discrimant

I am struggling to do this question Find the valus of $k$ so that $kx^2-2+kx+x=0$ has discriminant $Δ= -24$ I tried to use the formula $b^2-4ac$ but my answer is completely wrong. Any ideas ...
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$4ac-b^2\leq 3a(a+b+c)$ in quadratic

Suppose that the polynomial $(b+c)x^2+(a+c)x+(a+b)$ doesn't have real roots, where $a,b,c\in\mathbb{R}$. Prove that $4ac-b^2\leq 3a(a+b+c)$. The quadratic not having real roots means that $$(a+c)^2-4(...
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Discriminants of reduced polynomial equations

For reduced quadratic and cubic polynomial equations $$x^2 + px + q = 0$$ $$x^3 + px + q = 0$$ the discriminants seem closely related: $$D_2 = \Big(\frac{p}{2}\Big)^2 - \Big(\frac{q}{1}\Big)^1$$ $...
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Finding range of expression $f(x,y)=x^2+y^2$

Finding range of $f(x,y)=x^2+y^2$ subjected to the condition $2x^2+6xy+5y^2=1$ without Calculus Try: Let $k=x^2+y^2,$ Then $\displaystyle k=\frac{x^2+y^2}{2x^2+6xy+5y^2}$ Now put $\...
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Show that $Ax^2+Bx+C>0 $ for all real $x$ if and only if $A>0$ and $B^2-4AC<0$.

Show that For all real $x$ , $Ax^2+Bx+C>0 $ if and only if $A>0$ and $B^2-4AC<0$. Case 1 : Suppose that $A>0$ and $B^2-4AC<0$. Let $y=Ax^2+Bx+C$. Then $Ax^2+Bx+(C-y)=0$ ...
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Why is the discriminant of number fields greater 1?

Is there an immediate proof not using Minkowski's bound that we have $|\Delta_K|>1$ for all number fields $K \ne \mathbb Q$?
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Maximal extension of $\mathbb Q$ unramified outside a finite set of primes

I read the beginning of chapter 1 of Wiles paper on Modular elliptic curves and Fermat's Last Theorem. There it says: Let $p$ be an odd prime. Let $\Sigma$ be a finite set of primes including $p$ ...
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Are there only finitely many number fields with given prime factors of discriminant?

Let $S \subseteq \mathbb P$ be a finite set of prime numbers. Is it true that there are only finitely many number fields $K$ such that $p \mid \Delta_K$ implies $p \in S$?
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Investigating the solutions of $9^{x}+k(3^{x})+2=0$

I have worked this one through but still not 100% sure. the discriminant is $D=(k-2\sqrt{2})(k+2\sqrt{2})$. the quadratic equation gives $3^{x}=\dfrac{-k\pm\sqrt{k^2-8}}{2}$. as the RHS must be at ...
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Verification on “concurrent points”, and a cubic discriminant

It is well known that three lines are said to be concurrent precisely when they all meet at a point, namely the point of concurrency. In the paper On Sets Defining Few Ordinary Lines (v3) by Green &...
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Discriminant (in the context of PDE classification): $b^2 - 4ac$ or $b^2 - ac$?

I'm reading two textbooks on partial differential equations. In their respective sections on classification of PDEs (hyperbolic, parabolic, elliptical), they differ in what they describe as being the ...
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How many zeros does the polynomial have in the right half plane?

The polynomial is $f(z) = z^4+\sqrt{2}z^3+2z^2-5z+2$ If you check the image of the imaginary axis, you see that there are no zeros, so we can use the right semicircle from $iR$ to $-iR$,and make $R$ ...
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Find the least value of $n\in N$ for which $(n-2)x^2+8x+n+4>\arcsin(\sin12)+\arccos(\cos12)$ for every $x \in \mathbb {R}$

Find the least value of $n\in N$ for which $(n-2)x^2+8x+n+4>\arcsin({\sin12})+\arccos({\cos12})$ for every $x \in \mathbb {R}$ $(n-2)x^2+8x+n+4>\arcsin({\sin12})+\arccos({\cos12}) $ $(n-2)x^2+...
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To show: Galois group of a polynomial is not an alternating group .

I have a polynomial of this type: $p(t)+l(t)s(a,t)$ in $F_q(a)[t]$, where a=(a_0,...,a_m)$ are specialization. I could show that this polynomial is doubly transitive, by making $a_0=0$ (...
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Zagiers Class Number Definition of Binary Quadratic Forms

If I understood the definition of class numbers of discriminant $D$ for binary quadratic forms (b.q.f.) correctly, the class number $h(D)$ for a given discriminant $D$ is the number of equivalence ...
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Greatest common divisor of coefficients

I recently stumbled across the following statement. Let $d \in \mathbb{Z} \setminus \{0, 1\}$ be square free, $$ D = \begin{cases} \phantom{-} 4d & d \equiv 2, 3 \text{ mod } 4 \\[4pt] \,\,\...
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Connection between discriminant of ideal and quadratic number field.

Problem: Let $K$ be a quadratic field, $\mathcal{O}_K$ its ring of integers. Now I want to prove the equality $$D(\mathfrak{a}) = \mathcal{N}(\mathfrak{a})^2 \cdot \Delta_K $$ where $\mathfrak{a} \...
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Characterizing $f$ and $g$ such that $\deg(\gcd(f,g)) \geq 2$.

Let $f=f(t),g=g(t)\in \mathbb{C}[t]$, with $\deg(f),\deg(g) \geq 3$. A known result about the resultant of $f$ and $g$ says the following: The resultant of $f$ and $g$ is $0$ if and only if $f$ and $...
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Multivariable Resultant

I want to compute the resultant of $f_1,\cdots, f_m \in K[x_1,\cdots x_n]$ where $f_1, \cdots,f_m$ are forms of degrees $d_1,\cdots ,d_n$ respectively. Does anyone have any idea on how could I do this?...
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Does this fact about the minimal polynomial give an efficient diagonalizability criterion?

In a very nice paper "When Is a Linear Operator Diagonalizable?" by Marco Abate (Amer. Math. Monthly 104 (1997), 824-830) I found the following nice description of the minimal polynomial $\mu(T)$ of ...
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Properties of a polynomial with zero discriminant

In Wikipedia it says[1] that "The discriminant of a polynomial over an integral domain is zero if and only if the polynomial and its derivative have a non-constant common divisor." How does one ...
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discriminant of $x^p-1$

I am attempting to solve Artin 16.10.9, part (b). I have already solved (a). Let $f(x)=(x-α_1) \cdots (x-α_n)$. (a) Prove that the discriminant of $f$ is $\pm f'(α_1) \cdots f'(α_n)$, where $f'...
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Non zero discriminant

Let $p$ be a prime number. $k=\bar F_p$. Let $G=g(t)+F[t,x]\in k[t][x]$ be such that $G$ is irreducible in $x$ and separable in $t$, where $x$ is a polynomial in $F_{p}[t]$ of the form $f=\sum_{i=0}^{...
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Confusion about the definition of ideal (of ring of algebraic integers)

I am studying algebraic number theory and am confused about the following lemma. We prove that if $I \subset O_K$ a non-zero ideal then $$\textrm{disc}(I) = \textrm{disc}(O_K)\cdot N(I)$$ Then ...
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Matrix Notation Form of Roots of a Quadratic Equation

We know that the quadratic equation $$f(x)=ax^2+bx+c=0$$ has roots $$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}=-\frac b{2a}\pm \frac 1a\sqrt{-\left(ac-\frac {b^2}4\right)}$$ Also, $f(x)$ can be written in ...
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Adjoining square root of a discriminant

Let $f(X) = X^2 + bX + c \in K[X]$ be a quadratic polynomial whose discriminant is $\delta^2 = b^2 - 4c$. I understand that if $\alpha_1$ and $\alpha_2$ are the roots of $f$, then $\delta = \alpha_1 - ...
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Proving $a^4+b^4+c^4+(\sqrt {3}-1)(a^2 b c+a b^2 c+a b c^2 )\ge \sqrt {3} (a^3 b+b^3 c+c^3 a)$ for real $a$, $b$, $c$

If $a$, $b$, $c$ are real numbers, I have to prove: $$a^4+b^4+c^4+(\sqrt {3}-1)(a^2 b c+a b^2 c+a b c^2 )\ge \sqrt {3} (a^3 b+b^3 c+c^3 a)$$ Since $$a^4+b^4+c^4 \ge abc(a+b+c)$$ then it is enough ...
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Let $f \in F[x]$, then discriminant of $f$ lies in $F$.

Let $\alpha_1, \ldots, \alpha_n$ be the roots of $f$ in some splitting field $K$. Then define $disc_K(f) = \prod_{i<j} (\alpha_i - \alpha_j)^2$. Is it true $disc_K(f) \in F$? I used to prove $...
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Maximal Binary Quadratic Forms

I am currently working with positive-definite, reduced, primitive, integral binary quadratic forms, and I have noticed something interesting. Conjecture: Let $Q$ be a form of non-fundamental ...
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Writing the discriminant of an integer cubic polynomial with no double root as a combination of polynomials

Let $f(X)=X^3+aX^2+bX+c \in \mathbb Z[X]$ be a polynomial such that $f(X)$ and $f'(X)$ has no common root in $\mathbb C$. Let $\alpha_i$ , $i=1,2,3$ are the distinct roots of $f$ in $\mathbb C$. Let $...
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discriminant of a polynomial of degree 2 in 3 variables

Which is the correct way/ method/Formula to find the discriminant of a quadratic equation in 3 variables? Also, how to conclude that, whether this f is reducible or irreducible? Some one knows this ...
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Formula for discriminant of a polynomial of degree 2 in 3 variables

Which is the correct way/ method/Formula to find the discriminant of a quadratic equation $f$ in 3 variables? i.e., a quadratic form in 3 variables. Also, how to conclude that, whether this $f$ is ...
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Relation between conductor and discriminant of an elliptic curve

Let $E$ be an elliptic curve with complex multiplication and let $F=\mathrm{End}(E)\otimes \mathbb{Q}$. I'm looking for an (original) reference for the fact that discriminant of the elliptic curve $\...
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How can one prove that the quadratic equation $ax^2 + bx + c = 0$ with $a, b, c ∈ Z$ can't have 11 as discriminant?

How can one prove that the quadratic equation $ax^2 + bx + c = 0$ with $a, b, c \in \mathbb{Z}$ can't have 11 as discriminant?
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Relative discriminant of extension of number field

Let $K$ be a number field and let $L$ be an extension of $K$ with respective rings of integers $\mathcal{O}_L$ and $\mathcal{O}_K$. I am aware of something called the relative discriminant of $L$ ...
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$x^2+y^2=1$, find min/max of $(3x+2y)^2+(x+2y)^2$

I have been trying to solve this problem, For $x, y \in \mathbb{R}$ such that $x^2+y^2=1$, find the minimum and maximum value of $$(3x+2y)^2+(x+2y)^2$$ There are many ways to solve this problem ...
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671 views

Polynomial has real roots

I am trying to prove that the polynomial $f = X^3+pX+q \in \mathbb{R}[X]$ has three real roots iff $\Delta(f) = -(4p^3+27q^2) \geq 0$. I've figured out the left-to-right direction. Write $f = (X-\...
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Values of m for which the expression is positive

Q: Find the values of m for which the expression below is always positive. $x^2 + 2mx + (3m-2)$ I have attempted the question and know that I'm supposed to use the discriminant, however I'm having a ...
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Query Regarding Roots of Discriminant [closed]

Suppose I have an equation $x^2+kx+k$ and the required condition for its roots is $D=0$. After splitting the middle term we obtain $k(k-4)=0$. By the rules, the factors are $k=0,4$ but what if I ...