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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27 views

Discriminant of q-analogs of positive integers

Computations suggest that the discriminant of the polynomial $[n]_{q^m}=1+q^{m}+\dots+q^{(n-1)m}$ with respect to $q$ is $ \pm {\left( {m{{(mn)}^{n - 2}}} \right)^m}.$ Could you please give me a ...
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Suppose $a^2-4b$ $\neq 0$. Let $\alpha, \beta$ be the (distinct) roots of the polynomial $x^2+ax+b$. Then there is a real number $c$ s.t.

Claim: Let $a,b$$\in$$\mathbb{R}$ and suppose $a^2-4b$$\neq$$0$. Let $\alpha$ and $\beta$ be the (distinct) roots of the polynomial $x^2+ax+b$. Then there is a real number c such that either $\alpha$ $...
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Geometric interpretation of discriminant of a number field (and reference suggestion)

I recently finished a good chunk of Vakil's Algebraic Geometry book and am now brushing up on number theory with Milne's Algebraic Number Theory book, trying to relate things back to geometry when I ...
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How to prove that $\Pi_{1 \leq j \leq p^n, p \nmid j}((\zeta_{p^n}^j)^{p^{n-1}}-1) = p^{p^{n-1}}$?

How do I prove that $\Pi_{1 \leq j \leq p^n, p \nmid j}((\zeta_{p^n}^j)^{p^{n-1}}-1) = p^{p^{n-1}}$? I calculated some examples in Wolfram Alpha where the results were of this specific form, and it ...
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Knowing A and the discriminant, how to find B and C? [closed]

On a Quadratic equation of the form $ax^2+bx+c$, I am given the discriminant ($\Delta=b^2-4ac$) and A. To find B and C, I tried to solve it through $x1$ and $x2$...: $x1 = \frac{-b-\sqrt{\Delta}}{a}$ ...
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How are these two discriminants of polynomials related?

I was wondering the following problem, suppose we have $a,b$ such that $\mathbb Q(a)=\mathbb Q(b).$ Then consider the minimal polynomial of $a$, say $m_a(x)$ and $m_b(x)$ defined analogously. Is it ...
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Intuitive meaning of discriminant as determinant of trace values

Let $A$ be a commutative ring, $n\in\Bbb N$, $R$ be an $A$-algebra which is free $A$-module of rank $n$. If $x_i\in R$ for $i\lt n$, then the discriminant is defined as: $$\operatorname{dis}_{R/A}(x_i:...
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For an inner product space $V_K$, for which fields $K$ is the discriminant is a complete invariant of the form?

The discriminant is an invariant of a general symmetric bilinear form on a vector space $V$ over a field $K$. I'm trying to understand for which fields the discriminant is a complete invariant. First ...
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The discriminant as an invariant for the matrix representation of an inner product related to discriminant of quadratic formula i.e. $b^2-4ac$?

Let $V$ be a vector space equipped with the inner product $<-,->$. Let $A$ and $B$ be marix expressions for $<-,->$ in two different basis for $V$ that are equivalent, so that there exists ...
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Discriminant of a Quadratic Polynomial

I am a little confused about how the author came to this discriminant of this quadratic polynomial. I understand that the discriminant comes from solving for a variable, and the discriminant is ...
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Show this property of primitive quadratic forms

Let $f(x,y),g(x,y)$ be two primitive quadratic forms of discriminant $D < 0$. Then the following statements are equivalent: (i) $f(x,y)$ and $g(x,y)$ are in the same genus i.e. they represent the ...
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Discriminant of $\mathbb{Q}(\sqrt{-13})$

Here's my attempt at calculating the discriminant of $\mathbb{Q}(\sqrt{-13})$. The $\mathbb{Z}$-basis of its ring of integers $R=\mathbb{Z}(\sqrt{-13})$ is $\{e_1,e_2\}:=\{1,\sqrt{-13}\}$ Its complex ...
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Determine the sign of $169\cdot7-49\sqrt{123}$

Determine the sign of the discriminant of the equation $$49x^2-26\sqrt7x+\sqrt{123}=0.$$ The coefficients $a,b$ and $c$ of the equation are: $$a=49,\\b=-26\sqrt7\Rightarrow k=-13\sqrt7,\\c=\sqrt{123}.$...
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A Formula of Jet Bundles in Gelfand's Book

In Gelfand's book Discriminants, resultants, and multidimensional determinants he gives a formula: $$J_{1}(L)\cong J_{1}(\mathcal{O}_{X})\otimes L$$ Here $L$ is a line bundle on some complex variety. ...
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Where does the plus-minus come from in the quadratic formula? [closed]

In the formula $$x_{1,2} = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ for solving quadratic equations, where does the $\pm$ come from? The square root already results in both a positive and negative term, ...
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Discriminant of odd cyclic Galois extension is not a power of 2

Is there a way to prove that, given a Galois cyclic extension $\mathbb{Q} \subset F$ with odd order, there exists a prime $p \neq 2$ such that $p|\Delta_{F}$ ? I'm actually trying to prove that the ...
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381 views

Is there a simple criterion for when $e^{x}+ ax^{2}+ bx+ c$ has a zero?

Is there a simple criterion for when $$e^{x}+ ax^{2}+ bx+ c$$ has a zero ? Source: MathOverflow/@MattF. one simpler function $e^{x}+ ax+ b$ has a zero iff either or $- a+ a\!\ln\!\left ( -a \right )+ ...
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Residue field of splitting field is splitting field of residue field.

Let $f$ be an irreducible monic polynomial over $\Bbb Z$ and $L$ its splitting field. Let $p$ be a prime number and $\frak p$ be a prime ideal of ${\cal O} _L$ lying over $p$ (the question also makes ...
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Silverman exercise 3.1

I try to do exercise 3.1 from graduate elliptic curve by Silverman. I find a discussion here Silverman exercise 3.1 proving that two polynomials are relatively prime iff the discriminant is non-zero ...
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A good combination of an Olympiad inequality and its solution

Given three side-lengths $a, b, c$ of a triangle. Prove that $$a^{2}b\left ( a- b \right )+ b^{2}c\left ( b- c \right )+ c^{2}a\left ( c- a \right )\geq\left | \left ( b+ c- a \right )\left ( a- b \...
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The inverse type of Bernhard Leeb's solution for IMO‐1983–inequality

Given three side-lengths $a, b, c$ of a triangle. Prove that $$a^{2}b\left ( a- b \right )+ b^{2}c\left ( b- c \right )+ c^{2}a\left ( c- a \right )\geq 3\left ( a+ b- c \right )c\left ( a- b \right )\...
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Find $m\!\left(a,b,c\right)\!,n\!\left(a,b,c\right)\!\geq 0$ so $H:=m\!\left(a,b,c\right)\!-n\!\left(a,b,c\right)\!\left(a-b\right)\!\left(b-c\right)$

I'm doing research on creating an SOS new method, I need to the help. Introduction. Given three non-negative numbers $a,\!b,\!c,$ we have a symmetric polynomial $$a\left ( ab+ 1- b \right )\!\left ( ...
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If $L/K$ is an extension of number fields, show that the integer ${\rm disc}(K)$ divides ${\rm disc}(L)$.

If $L/K$ is an extension of number fields, show that the integer ${\rm disc}(K)$ divides ${\rm disc}(L)$. My attempt : We have a $\mathbb Z$-basis of $O_L$, then we can write down the basis elements ...
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To need a way of thinking about Ji_Chen's nice result $(a- c)^{2}+ (b- d)^{2}\geq\frac{7}{9}ab- \frac{7}{20}(c^{2}+ 4d^{2})$

given four real numbers $a, b, c, d$ Ji Chen gave a nice result on.AoPS $$\left ( a- c \right )^{2}+ \left ( b- d \right )^{2}\geq\frac{7}{9}ab- \frac{7}{20}\left ( c^{2}+ 4d^{2} \right )$$ The ...
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First-order recursion problem with a special cubic equation: by Ji_Chen

Given a natural number $n$ and a real number $d_{n},$ for the least $d_{n+1}$ so that $32d_{n}^{5}\!\left ( d_{n}+ 2 \right )\!=$ $$= 2\left ( 5d_{n}^{4}+ 8d_{n}^{2}+ 8d_{n}+ 8 \right )d_{n+ 1}^{3}+ ...
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377 views

A travelled inequality found by discriminant

Given three real numbers $x, y, z$ so that $1\leq x, y, z\leq 8.$ Prove that $$\sum\limits_{cyc}\frac{x}{y}\geq\sum\limits_{cyc}\frac{2x}{y+ z}$$ I found this inequality by discriminant, we realised ...
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Why is the discriminant of a quadratic what it is? I of course want to apply this to an nth degree polynomial.

Why do I keep getting that the discriminant of a quadratic is $4a^2c - ab^2$ when I use sylvester matrices? I know that I can factor out -a and get $-a(b^2-4ac)$. But this is going to be negative when ...
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In Fisher’s discriminant for multiple classes, How do you manage when $(Sw)$ is singular matrix (so you cant get $(Sw)^{-1}$)?

I am trying to use Fisher’s discriminant for multiple classes to reduce the Dimension of the MNIST data set, similar to this post: https://towardsdatascience.com/an-illustrative-introduction-to-...
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Upper bound for discriminant of Galois closure

In Lang's Algebraic Number Theory book he uses a certain bound on the discriminant of the Galois closure of a number field $K$ without proof stating that it is an easy exercise. Let $\tilde{K}$ be the ...
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Is there a notion of discriminant in general systems of nonlinear algebraic equations?

Given a system of nonlinear algebraic equations, is there a general way to define a discriminant? As far as I know you can define a discriminant in some cases, e.g. when you have elliptic equations ...
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102 views

Is there something similar to a discriminant for algebraic curves of the form $y^2 = f(x)$ (similar to the discriminant of elliptic curves)?

Consider an algebraic curve $X$ defined by an equation $y^2 = f(x)$ where $f$ is a polynomial of degree $d \geq 3$ with coefficients in a field $k$. For $m = 2$ and $d = 3$, we can get an equation of ...
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Controlling the discriminant size of a polynomial using roots

I'm trying to generate random monic irreducible polynomials in $\mathbb{C}[x]$ whose absolute discriminant is within a certain size range (say $10^t$, where $t$ is a positive integer). I also need to ...
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Discriminant of multivariable function

Hey I'm doing an online course and I just can't figure out what I'm doing wrong on one of the questions. The question is to find the discriminant of the function $$f(x,y) = 5x^2y^2 + 8x^2 + 9y^2$$ The ...
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How does the DISCRIMINANT really work?

I have the following equation (depended on param a) $(a^2-2a)x^2+2ax-1=0$ I want to find out what the behavior of this equation after changing the value (by ...
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Solving for an unknown value in inequalities

Q: Find the values of 𝑘 for which the quadratic expression 4𝑥^2 + 12𝑘𝑥 − 𝑘 is always positive. What I’ve tried: The discriminant is > 0 therefore b^2-4ac>0 a=4 b=12k c=-k This resulted in: ...
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Given two natural numbers $a,b$ so that $ab+\left(a+1\right)\left(b+1\right)=2n=fixed,$ find $a\left(a\leq b\right)$ such that $b-a$ at the least.

Problem. Given two natural numbers $a, b$ so that $ab+ \left ( a+ 1 \right )\left ( b+ 1 \right )= 2n= fixed,$ by programmin' find $a\left ( a\leq b \right )$ such that $b- a$ at the least. I haven't ...
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What is the geometric intuition for the Sylvester matrix?

Suppose we have two polynomials $p(x)$ and $q(x)$ with degrees $m$ and $n$, respectively, and we consider $S^T$: the transpose of the $(m+n)\times(m+n)$ Sylvester matrix associated to these ...
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Alternate way to solve this algebra problem?

Working on a problem that says given the absolute value of the difference of the roots of $ax^2 + bx + c$ as $2$, what is the absolute value of the difference of the roots of $ax^2 + 6bx + 36c$? I ...
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Calculate explicit form for parameters of LDA

In a Linear Discriminant Analysis (with two classes $A, B$) we use a decision rule of the form $$\hat y = \begin{cases} A & \text{if } \text{sign}(\hat w^t x + \hat c) > 0 \\ B & \...
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Range of quadratic function using discriminant

Let $x^2-2xy-3y^2=4$. Then find the range of $2x^2-2xy+y^2$. Let $2x^2-2xy+y^2=a$. Then $ax^2-2axy-3ay^2=4a=8x^2-8xy+4y^2\implies (a-8)x^2-(2a-8)xy-(3a+4)y^2=0$. We divide both side by $y^2$ and let $...
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How can I obtain decision boundaries for four normal distributions?

I have plotted four Gaussian distributions in a 3D surface, as shown in the image. I aim to plot decision boundaries, in a contour plot, for these distributions. I have discriminant functions of each ...
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1answer
305 views

Understanding the term “double roots” of a quadratic equation [duplicate]

We know that when a quadratic equation $ ax^2 + bx + c = 0 $ has zero discriminant value, then the quadratic equation has only "one root". But why do some mathematician call it "double ...
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Finding the formula for the discriminate of an algebraic integer

Let $n \geq 2$, and $\alpha$ be an algebraic integer with minimal polynomial $$f(x) = x^n + ax + b, \, \, \, a, b \in \mathbb{Z}.$$ Show that $$\Delta(\alpha) = (-1)^{\frac{n(n-1)}{2}}((-1)^{n-1}(n - ...
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61 views

Discriminant of integral basis

Let $A=\mathbb{Z}\alpha_1+...+\mathbb{Z}\alpha_n$, $B=\mathbb{Z}\beta_1+...+\mathbb{Z}\beta_n$, be two lattices, s.t. $A\subseteq B$. Then $d(\alpha_1,...,\alpha_n)=c^2d(\beta_1,...,\beta_n)$, where $...
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39 views

Discriminant of the minimal polynomial in simple extension

Suppose that $A\subset B$ is an integral extension of domains and $A$ is integrally closed. Also assume that, if $F_A$ and $F_B$ the quotient fields of $A$ and $B$ then $F_B$ is a simple extension of $...
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116 views

Why are the roots of this recursive defined polynomial bound by the roots of the discriminant of the characteristic polynomial?

Let's define a polynomial recursively as: $$ \begin{align} p_0(x) &= 1 \\ p_n(x) &= x \sum\limits_{k=1}^n a_k p_{n-k}(x) \end{align} $$ Let $a_k$ be an arithmetic progression. Question: Why ...
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What does this footnote on finding the range of a rational function mean?

The book I am currently studying for rational functions states that To find the values the function $y=\frac{1}{x^2-3x+2}$ can not take we can rearrange to give $$yx^2-3yx-(2y-1)=0$$ And by using the ...
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1answer
87 views

How do you calculate discriminants of higher degree polynomials?

I have a degree 6 polynomial with coefficients in terms of $a$ that I want to calculate the discriminant of. Is there an easy way to do this like an online calculator? The issue I am having is that ...
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38 views

Finding $m$ given that $(m^2 - 1)x^2 - 3(3m - 1)x + 18 = 0$

(a) Suppose $m$ is an integer so that $(m^2 - 1)x^2 - 3(3m - 1)x + 18 = 0$ has two positive integer roots. Find $m.$ (b) Now, suppose that we have a triangle $ABC$ with sides $a,b,c$ such that \begin{...
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2answers
89 views

Finding $m$ given that $9m^2 + 25m + 26$ is a product of consecutive integers

Suppose $m$ is an integer such that $9m^2 + 25m + 26$ is the product of two consecutive integers. Find $m.$ I first let $k$ be equal to the larger of the two consecutive integers so that I can set up ...

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