# Questions tagged [discriminant]

The discriminant tag has no usage guidance.

27
questions

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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 ...

**0**

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27 views

### Fixing coefficients using analytic structure

I am trying to understand an exercise in a set of lecture notes on random matrices - Eynard - Random matrices - given in the paragraph following (4.6.60) (pp. 70–71).
Specifically, I am trying to fix ...

**4**

votes

**1**answer

200 views

### 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(\...

**9**

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187 views

### 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 ...

**7**

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188 views

### 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 ...

**8**

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255 views

### 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 ...

**8**

votes

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137 views

### 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)|}...

**3**

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101 views

### 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 $\...

**2**

votes

**1**answer

178 views

### 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}...

**1**

vote

**2**answers

113 views

### 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 $\...

**2**

votes

**1**answer

216 views

### 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 ...

**6**

votes

**4**answers

611 views

### 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 $...

**1**

vote

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63 views

### 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 ...

**1**

vote

**2**answers

214 views

### 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}...

**4**

votes

**1**answer

241 views

### 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{...

**9**

votes

**0**answers

212 views

### 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 ...

**3**

votes

**0**answers

80 views

### 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 ...

**4**

votes

**1**answer

199 views

### 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 ...

**13**

votes

**1**answer

754 views

### 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 ...

**0**

votes

**1**answer

1k views

### 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 ...

**3**

votes

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172 views

### 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 ...

**2**

votes

**0**answers

126 views

### 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 ...

**1**

vote

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847 views

### 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 ...

**5**

votes

**1**answer

263 views

### 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 ...

**3**

votes

**1**answer

281 views

### 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 ...

**8**

votes

**2**answers

804 views

### 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 ...

**5**

votes

**1**answer

310 views

### 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" ...