# Questions tagged [resultant]

This tag is for the resultant of polynomials, which detects when two polynomials have a common factor.

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### Algorithm for the resultant of two bivariate polynomials

I'd like to know how to calculate the resultant of two bivariate polynomials. Searching through the web, I read that there is an Euclid-like algorithm for univariate polynomials, and that one can use ...
• 2,012
1 vote
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### if $\exists$ nontrivial polynomial relation $s(X) \cdot A(X) + t(X) \cdot B(X) = 0$, then do $A$ and $B$ have a common factor?

let $R$ be a commutative ring with unit, and fix elements $A(X)$ and $B(X)$, not both zero, in $R[X]$. suppose that there exist polynomials $s(X)$ and $t(X)$ in $R[X]$, respectively of degrees less ...
• 601
1 vote
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### Calculate the discriminant of $X^n+aX^{n-1}+b$, given the discriminant of $X^n+aX+b$

Calculate the discriminant of $X^n+aX^{n-1}+b \in \mathbb{Z}[X]$, knowing that the discriminant of $X^n+aX+b$ is $(-1)^{\frac{n^2+n-2}{2}}(n-1)^{n-1}a^n+(-1)^{\frac{n(n-1)}{2}}n^nb^{n-1}$. All I have ...
• 1,342
1 vote
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• 7,065
1 vote
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### Resultant Understanding

The resultant of two polynomials $f\left(x\right)=a_{n}x^{n}+\ldots+a_{0},g\left(x\right)=b_{n}x^{m}+\ldots+b_{0}\in K\left[x\right]$ over a field K defined as $\det R\left(f,g\right)$ when $R(f,g)$ ...
• 39
1 vote
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• 57
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### A vector represents a force with its magnitude and direction but could it also represent the time it was applied for?

I'm an A2 igsce math student and I'm taking mechanics for the first time in math this session. I know that what I'm going to say is wrong judging from my teacher's reaction when I asked him about this ...
• 385
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### What is the geometric intuition for the Sylvester matrix?

Suppose we have two polynomials $p(x)$ and $q(x)$ with degrees $m$ and $n$, respectively, and we consider $S^T$: the transpose of the $(m+n)\times(m+n)$ Sylvester matrix associated to these ...
173 views

### Alternative proof for multiplicativity of resultant?

The formula $R(fg,h)=R(f,h)R(g,h)$ follows easily if you express each resultant in terms of the roots of the polynomials involved. It can also be obtained by relating $R(f,h)$ to the determinant of ...
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### Merging polynomials together

Given the (monic) polynomials $P$ and $Q$, we can split them over an appropriate field into linear factors to write $P(x)=(x-a_1)\dotsb(x-a_m)$ and $Q(x)=(x-b_1)\dotsb(x-b_n)$, and then form the ...
• 4,649
1 vote
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### rational function with distinct roots

Let $r(X)$ be a rational function over $\mathbb{C}$. Is there (always) a complex number $z$ such that the rational function $r(X)-r(X+z)$ has distinct roots?
• 11
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### Reference request for the Elimination Properties of Resultants

Let $f,g$ be polynomials in $k[y_1,\dots,y_n][x]$ over a field $k$. Assume that at least one of $f$ and $g$ is of positive degree in $x$. Denote by $\operatorname{res}_x(f,g)$ the resultant of $f$ and ...
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1 vote
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• 6,595
1 vote
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### Where are Kronecker's lectures on polynomial resultants?

From Mathworld, which in turn cites Henry Fine's biography here, states: Kronecker gave a series of lectures on resultants during the summer of 1885. Though a search for lectures by Kronecker on ...
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