Questions tagged [discriminant]

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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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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
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1answer
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(\...
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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 ...
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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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0answers
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
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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)|}...
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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
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1answer
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}...
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2answers
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
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1answer
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 ...
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4answers
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 $...
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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 ...
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2answers
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
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1answer
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{...
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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
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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
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1answer
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 ...
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1answer
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 ...
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1answer
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
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0answers
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 ...
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0answers
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 ...
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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
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1answer
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
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1answer
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
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2answers
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
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1answer
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" ...