# 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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### Relating discriminants of hyperelliptic curves to discriminants of their defining polynomials

Let $C$ be a hyperelliptic curve defined by an equation of the form $$C: y^2=f(x)$$ where $f$ is a polynomial of prime degree $p\geq3$, over a complete field $K$ of residue characteristic $p$. ...
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### discriminant to distinguish parallel line and double line degenerate conic sections

A real affine conic section is the zero locus in $\mathbb{R}^2$ of the quadratic form $$q(x,y)=ax^2+2bxy+cy^2+2dx+2ey+f=0.$$ We may understand this as the $Z=1$ affine patch of the locus in the ...
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### Finding the discriminant of a quaternion algebra

Consider the totally real number field $F=\mathbb{Q}(\zeta_{10}+\zeta_{10}^*)$. Consider the quaternion algebra $Q=(\frac{-1,-1}{F})$. How do I compute the discriminant of this algebra? I gave ...
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### convert a 1-dimensional set of points to a 2-dimensional parabola with explicit embedding

I am trying to rephrase to better understand concepts regarding discriminant functions for classification using explicit embedding. I report a very easy diagram found online that from 1-dimension ...
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### Smallest possible value of $k$ such that the roots of $x^2-127x+k=0$ are positive integers [closed]
In a triangle, two sides have equal lengths both shorter than the third side. The length of the three sides are all integers and all satisfy the equation $x^2-127x+k=0$, $k$ is a constant. Find the ...