# Questions tagged [discriminant]

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### Distinguishing between prime factors of cubic discriminant and polynomial discriminant

Let $f(x)\in\mathbb{Q}[x]$ be an irrreducible cubic with root $\alpha$. Let $K=\mathbb{Q}(\alpha)$. There may be primes dividing $\text{disc}(f)$ that don't divide $\operatorname{disc}(K)$, so an ...
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### Non-isomorphic cubic fields with a given discriminant

For a cubic field $K$ with defining polynomial $P(x)=x^3 + \frac{39}{25}x^2 + \frac{22}{25}x +\frac{4}{25}$ Magma calculates the discriminant $D=-3340$. ...
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### Reference request: Discriminant of a $V_4$-extension of local fields is the product of discriminants of intermediate fields

Disclaimer - cross-posting: I already posted this question on MSE, here. In line with the accepted answer of this meta question, I am also asking it here, since it is a research-level question and it ...
1 vote
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### Proving Geometric Inequality Using Equation Discriminant

I met this question before: An acute $\triangle ABC$ (you can imagine $BC$ below) has a point $D$ on side $AC$. The line parallel to BC through $D$ meets $AB$ at $E$, and the parallel line $BD$ ...
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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 $f = \sum_{i=0}^n f_iy^i \in \mathbb{C}[x,y]$ be a polynomial (where $f_i \in \mathbb{C}[x]$ with $f_0,f_n \ne 0$)...
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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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1 vote
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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 $\... • 17.7k 2 votes 1 answer 461 views ### Is the set of integers represented by a quadratic form of non-fundamental discriminant a subset of the rep. set of a form of fundamental discriminant? I am currently working with positive-definite, reduced, primitive, integral binary quadratic forms, and I have noticed something interesting. Conjecture: Let$Q$be a form of non-fundamental ... 6 votes 4 answers 663 views ### What is the essence of the constant factor in the standard definitions of the discriminant? Let$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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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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### 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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1 vote
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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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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" ...