Questions tagged [discriminant]
The discriminant tag has no usage guidance.
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Possible integer values of discriminants of irreducible integer-valued polynomials
Background:
The discriminant of the constant polynomials is undefined.
Every linear polynomial has discriminant $1$.
No integer-valued polynomial with no rational non-constant square factor can have ...
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Construction of a symmetric polynomial in the roots that acts like the discriminant
The discriminant $\Delta(P)$ of a monic polynomial $P(x)=x^n + a_{n-1} x^{n-1} + \dotsb + a_0$ of degree $n$, when expanded (using elementary symmetric polynomials), is a symmetric polynomial of ...
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Does a discriminant condition on $f(x,y)$ imply that $f$ is weighted homogeneous?
[This is an updated version of https://math.stackexchange.com/questions/4522399/.]
Let $f = \sum_{i=m}^n f_iy^i \in \mathbb{C}[x,y]$ be a polynomial (where $f_i \in \mathbb{C}[x]$ with $f_m,f_n \ne 0$ ...
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Number fields with given discriminant
In the special case of number fields that are the splitting fields of irreducible polynomials of degree 5 or 6 with Symmetric Galois group (so degree 120 or 720 over Q), is there a good upper bound on ...
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Why the sign in the definition of the discriminant?
Consider the split monic $f=\prod_{i=1}^n(x-x_i)\in \mathbb Z[x_1 ,\dots ,x_n,x]$. Its discriminant is usually defined as $$(-1)^{n(n-1)/2}\prod_{i=1}^nf^\prime(x_i)=\prod_{1\leq i<j\leq n}(x_i-x_j)...
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Discriminants of some $q$-analogs of $(1+x)^n$
Let $[n]_q=1+q+\dots +q^{n-1}$, $ {[n]_q}! =[1]_q [2]_q \dots [n]_q$ and $\binom{n}{j}_q = \frac{[n]_q!}{[j]_q![n-j]_q!}$ be the usual $q$-notation.
Consider the polynomials $p_n(q,r,x)= \sum_{j=0}^n ...
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Does there exist a type of discriminant not only for irreducible polynomials but also for exponential functions, logarithm functions?
I think discriminant is the strongest tool that I've used_ https://math.stackexchange.com/q/4035405/822157, however, does there exist a type of discriminant not only for irreducible polynomials but ...
user164469
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A mysterious expression from a discriminant
I recently investigated $\textrm{Discriminant}_u(\omega P(u) +(z-u)P'(u))$, where $P(u) := u^3 + au + b$ and $\omega$ is a real parameter (with $\omega\in(0,1)\cup(1,3)$) associated with the order of ...
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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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Discriminant of $\alpha P(u) + (z-u) P'(u)$
I'm trying to find a “closed form” of $\textrm{Discriminant}_u(f(u))$, where $f(u) := \alpha P(u) + (z-u) P'(u)$.
Here $P(u)$ is a monic polynomial of degree $d > 1$ with $u\in\mathbb{C}$, $\alpha$ ...
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What is the subgroup of $\mathrm{SL}(n,\mathbb{C})$ which preserves the discriminant?
$\DeclareMathOperator{\SL}{\operatorname{SL}}$Let $\mathcal{P}_{n-1}$ be the space of complex polynomials in one variable, say $z$, of degree at most $n-1$. As a complex vector space, it is clearly $n$...
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Discriminants of Gleason's period-$n$ polynomials for the Mandelbrot set
Gleason's polynomials are the sequence of monic integer polynomials defined recursively by
$$
\prod_{d \mid n} G_d(c) = (((c^2+c)^2+c)^2+\cdots+c)^2+c \quad \quad \quad [\textrm{$n$ iterates}],
$$
for ...
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Discriminant mod 8
Let $f,g\in\mathbb{Z}[X]$ be monic polynomials of degree $n$, with discriminants $\Delta_{f}$ and $\Delta_{g}$, such that $f=g$ in $\mathbb{F}_{2}[X]$ and $\Delta_{f}$ is odd.
Is there an elementary ...
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Discriminant of a radical extension of a quadratic number field
Let $K=\mathbb Q(\sqrt 5)$ and $\varepsilon = \frac{3 + \sqrt 5}{2}$ its totally positive fundamental unit (i.e. it generates the subgroup of totally positive units). For any $n \geq 3$, let $L_n = K(\...
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How small may the discriminant of an $S_d$-field be?
In every degree $d$, the Galois closure of the typical number field has the maximal possible Galois group $S_d$. Denote by $f(d)$ the least absolute value of a discriminant of an $S_d$-field of degree ...
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Number fields ordered by discriminant
Since the discriminant of a number field $K \neq \mathbb{Q}$ is bounded from below by an exponential of the degree $[K:\mathbb{Q}]$, for instance by Minkowski's Geometry of Numbers bound, there are ...
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Is this a possible strengthening of the Lehmer conjecture?
Here is another possible refinement of the Lehmer conjecture.
For $\alpha \in \overline{\mathbb{Q}}^{\times}$, let $C_{\alpha} \subseteq \mathbb{Q}(\alpha)$ be the maximal cyclotomic field contained ...
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Is there an approximate formula for the discriminant of a sparse polynomial?
Consider integer polynomials $P \in \mathbb{Z}[X] \setminus \{0\}$ of a degree $D \geq 1$ and without multiple complex roots. Let me introduce a notation
$$
d(P) := \frac{1}{D} \log{|\mathrm{Disc}(P)|}...
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When does a number field have $p$-rank greater than $n$?
Consider $F/\mathbb{Q}$, a number field. Let $S$ be a finite set of primes of $F$ containing the Archimedean primes. Let $n$ be any natural number and $L_n$ be a finite extension of $F$ such that $\...
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Definition of a Discriminant in Three Variables
I am studying pell conics and the source I am using (Franz Lemmermeyer: Conics - A Poor Man's Elliptic Curves) defines its discriminant as follows:
For equations of the form $X^2 + XY + \frac{1-d}{4}...
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What would be a standard reference for the formula of the discriminant of $f(t^d)$?
I've posted this to Math.SE about a month ago:
Seems like
$$
\Delta(a_0+a_1t^d+a_2t^{2d}+...+a_nt^{nd})=(-1)^{n\frac{d(d-1)}2}d^{nd}(a_0a_n)^{d-1}[\Delta(a_0+a_1t+a_2t^2+...+a_nt^n)]^d,
$$
where $\...
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Is the set of integers represented by a quadratic form of non-fundamental discriminant a subset of the rep. set of a form of fundamental discriminant?
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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What is the essence of the constant factor in the standard definitions of the discriminant?
Let $f(x) = x^m+\sum_{j=0}^{m-1}f_{m-j}x^j\in P[x]$ be a monic polynomial over a field $P$ and let $f(x) = (x-\alpha_1)\cdot\ldots\cdot(x-\alpha_m)$ be a factorization of $f$ over an extension field $...
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Reducible binary quadratic form
Let $f(x,y)=(ex+fy)(gx+hy); \ x,y,e,f,g,h \in \mathbb{Z}$ be a reducible integral binary quadratic form. Is there a criterion to determine if a number is represented by this form? In particular, does ...
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Show that $\sum_{i=0}^{\frac{p-1}{2}} {{\frac{p-1}{2}}\choose {i}}^2 x^{\frac{p-1}{2}-i}$ is separable
This is a problem occurs in my research. For any algebraically closed field $k$ of characteristic $p$. I want to show that $\sum_{i=0}^{\frac{p-1}{2}} {{\frac{p-1}{2}}\choose {i}}^2 x^{\frac{p-1}{2}-i}...
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A generalization of the discriminant of a polynomial
Let $\mathbb{K}$ be a field and let $f \in \mathbb{K}[x]$ be a monic polynomial of degree $n$. Suppose that $\alpha_1, \ldots, \alpha_n$ are all the roots of $f$ (in some algebraic closure of $\mathbb{...
user40023
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Is the discriminant of a free (as a module) $R$-algebra always congruent to a square modulo 4?
Let $R$ be a commutative ring. Let $A$ be an $R$-algebra (i.e., an $R$-module
equipped with an $R$-bilinear multiplication map that turns $A$ into a unital
ring). We do not require $A$ to be ...
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Does anyone know anything about the 2-valuation of the discriminant of a polynomial?
Take a random polynomial $f$ with integer coefficients (e.g., choose coefficients between $1$ to $B$ of a fixed degree $n$ and let $B$ tend to $\infty$). Using computer we noted that the 2-valuation ...
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Principal Minors of the Resultant
Let $x_1, \ldots, x_n$ be variables, $e_n$ be the elementary symmetric polynomials. I will denote the discriminant by
$$D_n(x_1, \ldots, x_n) = \prod_{i<j} (x_i - x_j)^2$$
And a generalized ...
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Most discriminants are almost squarefree
Write, for $f(x) = x^d + a_2 x^{d-2} + \cdots + a_d\in \mathbb{Z}[x]$, $H(f) := \max(|a_i|^{\frac{1}{i}})$.
Does anyone know of a reference that would allow me to show that the proportion of $f$ with ...
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Discriminant of a polynomial in two variables
I want to compute the discriminant of the following polynomial
$$
F(X,Y)=X^mY^n+\sum_{i=0}^{m-1}\sum_{j=0}^{n-1}c_{ij}X^iY^j.
$$
Here the discriminate means the equation $D(c_{i,j})$ in the variables ...
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Discriminant polynomial generalizing the usual discriminant
I wonder if anybody has seen the following natural polynomial.
Given a monic univariate polynomial $P(z)$ of degree $N$, denote its roots by
$z_1,..., z_N$. Now form a new polynomial $Q(z)$ of ...
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Genus 2 hyperelliptic cryptography : typical discriminant and class number
As far as I know, there is no standard yet for cryptography based on the DLP over Jacobians of genus 2 curves. Yet, what can we say about the class number, and the discriminant of the complex ...
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Discriminant of a compositum of number fields, a bound?
Given two number fields $E$ and $F$, is there a bound on $|d_{EF}|$, the absolute value of the absolute discriminant of the compositum of fields $EF$, in terms of $d_E$, $d_F$, and the extension ...
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Discriminants of Clifford algebras
I have a Clifford algebra defined over a field of characteristic not equal to $2$. Is there a formula for its discriminant in terms of the corresponding symmetric bilinear form (or in terms of its ...
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plane cubics and conic bundles
It is well known that any plane cubic curve can be obtained as the discriminant locus of a conic bundle (actually even just of a net of conics). Does this hold true also for all nodal cubics (with ...
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Parameter space for complete intersections and their discriminant
Consider globally complete intersections in $\mathbb{P}^n$, of codimension $k$, of some fixed multi-degree $(d_1,\dots,d_k)$.
Is there some nice (i.e. "explicit") parameter space for them?
(even if ...
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Prime-like elements of rings
An element $p$ of a commutative ring $R$ is called "prime" if, for any $a,b\in R$, whenever $ab$ is a multiple of $p$, either $a$ or $b$ is a multiple of $p$.
Is there a word for the "prime-like" ...
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