# Questions tagged [resultant]

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

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### $f,g \in k[t]$ satisfying several conditions

Let $f=f(t), g=g(t) \in k[t]$, $k$ is a field of characteristic zero. Assume that: (i) $k(f,g)=k(t)$. (ii) $k(f',g')=k(t)$. (iii) $\langle f'',g'' \rangle = k[t]$. Is it true that (iv) $k[f',g']=k[t]$...
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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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### Which relations do imply the fact that two polynomials have at least two common roots?

When two polynomials $f(x)$ and $g(x)$ have a common root, then $\mathrm{Res}(f,g) = 0$ for their resultant. It looks like we should have at least two relations when we know that $f(x)$ and $g(x)$ ...
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### An unjustified bound in calculation of polynomial resultants

In an endeavor to prove Bezout's theorem for algebraic plane curves, I am finding myself stuck on one particular result that does not have a good reference in any literature. I am trying to prove weak ...
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### resultant of a binomial and a trinomial

Does anyone know of any papers dealing with the resultant of a binomial $x^n+a$ and a trinomial $x^r+bx^s+c$? Even special cases would be of interest. The resultant of two binomials is well known, ...
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### contributions of Riemann Hypothesis to physics if the Riemann zeta function is a solution for known differential equation? [closed]

There are several consequences of the Riemann hypothesis in many area as Number theory , complex analysis $\cdots$ ,I'm interesting to know what about those consequences if the Riemann zeta function ...
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### Bivariate polynomials coprime implies finitely many common roots

Let $f,g \in \mathbb{Q}[x,y]$ be coprime. I want to show that there are only finitely many common roots, i.e. only finitely many pairs $(a,b) \in \mathbb{C}^2$ with $f(a,b)=g(a,b)=0$. As a hint I ...
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### Hidden variable resultant for solution of a system of polynomial equations

I am trying to implement a method from as paper that solves a practical problem. The core part is solving of a system of polynomial equations of same total degree. We have a system of 10 polynomial ...
### Finding the minimal polynomial of $uv$, when those of $u$ and $v$ are given and of degree three
Let $k \subseteq L$ be a finite separable field extension (not necessarily Galois) of degree $3$, and assume that $L=k(u)=k(v)$, for some $u,v \in L$. Clearly, $k \subseteq k(uv) \subseteq k(u)=k(v)$, ...