This tag is for the resultant of polynomials, which detects when two polynomials have a common factor.
Questions tagged [resultant]
111 questions
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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 $g$ with respect to $x$.
On the Wikipedia page…
Randy Marsh
- 3,786
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2 answers
Algebraic relation between polynomials
The problem statement is "Let $F \in \mathbb{C}[t]$ have degree at most $D \geq 1$, and let $G \in \mathbb{C}[t]$ have degree $E \geq 1$.
Show that there is a $P \not = 0$ in $\mathbb{C}[X,Y]$ with degree at most $E$ in $X$ and $D$ in $Y$ such that…
user413766
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Are there any examples of math olympiad problems that can be solved by modern math?
I am looking for word problems that can be tackled by subjects like category theory, commutative algebra, nonlinear algebra, algebraic geometry etc.
5
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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 with those polynomials. The determinant of $S^T$ is the resultant of…
Aaron Cao
- 75
4
votes
2 answers
Solving system of non-linear equations
Many sources states that a system of equations could be solved by polynomials with degrees as worst as the product of degrees of the original SOE, but I wonder if this claim ever has a non-trivial numerical example.
Take for example, the system of…
Thinh Dinh
- 8,233
4
votes
1 answer
Sylvester matrix without coordinates and its geometry
The (transpose of) the Sylvester matrix of two polynomials $f,g\in A[x]$ represents the following $A$-linear morphism w.r.t the monomial bases of all $A$-algebras involved.
$$\tfrac{A[x]}{\langle g\rangle}\oplus\tfrac{A[x]}{\langle…
Arrow
- 14,390
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Alternative proof for multiplicativity of resultant?
The formula $R(fg,h)=R(f,h)R(g,h)$ follows easily if you express each resultant in terms of the roots of the polynomials involved. It can also be obtained by relating $R(f,h)$ to the determinant of multiplication by $f$ on the space $K[x]/(h)$. But…
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Concerning the subalgebra generated by two elements
Let $f=f(t),g=g(t) \in \mathbb{C}[t]$ be two separable polynomials of degrees $\deg(f)=n \geq 2$ and $\deg(g)=m \geq 2$, namely, $f$ has $n$ distinct roots and $g$ has $m$ distinct roots.
Denote $d=d(t)=\gcd(f(t),g(t))$, and assume that $\deg(d)=l…
user237522
- 7,263
4
votes
2 answers
Simultaneous real solution of $x^3+y^3+1+6xy=0$ & $xy^2+y+x^2=0$
I am trying to solve the following system of non-linear equations in real numbers:
$x^3+y^3+1+6xy=0$ & $xy^2+y+x^2=0$, with $x,y$ real.
I can only see that $xy\ne 0$.
I have no clue whether a solution exists or not and how to find any solution. I…
user521337
- 3,735
4
votes
2 answers
The resultant/ discriminant of a polynomial in one variable is not zero
If the resultant or discriminant of a polynomial is not zero, can we conclude critical points are distinct?
Math123
- 1,263
4
votes
1 answer
Applying the quadratic Tschirnhausen transformation
As per my previous question, I attempted to take dxiv's approach, though I can't seem to make much headway. Considering the simpler problem $x^3=x+a$ and the substitution $y=x^2+mx+n$, I got the…
Simply Beautiful Art
- 76,603
4
votes
2 answers
How to remove the second two leading terms in the general quintic with just algebra?
Motivated by How to transform a general higher degree five or higher equation to normal form?
The goal of the linked question is to transform the general quintic
$$x^5+ax^4+bx^3+cx^2+dx+e=0$$
into Bring-Jerrard normal form.
Tito Piezas III begins…
Simply Beautiful Art
- 76,603
4
votes
1 answer
Resultant contains all common roots as linear factors?
Let $f,g \in \mathbb{C}[x,y,z]$ be homogeneous polynomials, so they define projective plane curves $C$ and $D$ in $\mathbb{C}P^2$. We are interested in Bezout's theorem applied to $C \cap D$. Write $f$ and $g$ as polynomials in $z$:
$$
f(x,y,z) =…
Herng Yi
- 3,236
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0 answers
Resultant of two special trinomials
Consider $f(x)=x^n-x^s-1$ and $g(x)=x^i-x^j-1$ , I want to find $Resultant(f,g)$. It is well known that it is determinant of a Sylvester matrix but, I am finding it to obscure to evaluate in that way. Is there some known result for such special…
xyz
- 919
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1 answer
What are the algorithms to compute the resultant ${\rm Res}(f(x),g(x),x)$ of two univariate polynomials $f,g$?
I am trying to implement an algorithm for computing Res(f(x),g(x),x) where f(x) and g(x) uni variate polynomials with integer coefficients. Could any one list the various algorithms for computing Res(f(x),g(x),x) along with a brief comparison (e.g.…
user2912941
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