This tag is for the resultant of polynomials, which detects when two polynomials have a common factor.
Questions tagged [resultant]
116 questions
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Why is $\operatorname{res}(fg,h)=\operatorname{res}(f,h)\cdot\operatorname{res}(g,h)$, where $\operatorname{res}$ stands for resultant?
I'm learning Computer Algebra and met an exercise asking me to prove that $$ \operatorname{res}(fg,h)=\operatorname{res}(f,h)\cdot\operatorname{res}(g,h) $$ where $f(x)$, $g(x)$ and $h(x)$ are polynomials and $\operatorname{res}$ stand for…
Roun
- 3,237
6
votes
1 answer
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…
Randy Marsh
- 3,836
6
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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…
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…
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
- 10,088
4
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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. $$\tfrac{A[x]}{\langle g\rangle}\oplus\tfrac{A[x]}{\langle…
Arrow
- 14,560
4
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Alternative proof for multiplicativity of resultant?
The formula $R(fg,h)=R(f,h)R(g,h)$ follows easily if you express each resultant in terms of the roots of the polynomials involved. It can also be obtained by relating $R(f,h)$ to the determinant of multiplication by $f$ on the space $K[x]/(h)$ . But…
4
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Concerning the subalgebra generated by two elements
Let $f=f(t),g=g(t) \in \mathbb{C}[t]$ be two separable polynomials of degrees $\deg(f)=n \geq 2$ and $\deg(g)=m \geq 2$ , namely, $f$ has $n$ distinct roots and $g$ has $m$ distinct roots. Denote $d=d(t)=\gcd(f(t),g(t))$ , and assume that $\deg(d)=l…
user237522
- 7,267
4
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2 answers
Simultaneous real solution of $x^3+y^3+1+6xy=0$ & $xy^2+y+x^2=0$
I am trying to solve the following system of non-linear equations in real numbers: $x^3+y^3+1+6xy=0$ & $xy^2+y+x^2=0$ , with $x,y$ real. I can only see that $xy\ne 0$ . I have no clue whether a solution exists or not and how to find any solution. I…
user521337
- 3,755
4
votes
2 answers
The resultant/ discriminant of a polynomial in one variable is not zero
If the resultant or discriminant of a polynomial is not zero, can we conclude critical points are distinct?
Math123
- 1,283
4
votes
1 answer
Applying the quadratic Tschirnhausen transformation
As per my previous question , I attempted to take dxiv's approach, though I can't seem to make much headway. Considering the simpler problem $x^3=x+a$ and the substitution $y=x^2+mx+n$, I got the following:…
Simply Beautiful Art
- 77,026
4
votes
2 answers
How to remove the second two leading terms in the general quintic with just algebra?
Motivated by How to transform a general higher degree five or higher equation to normal form? The goal of the linked question is to transform the general quintic $$x^5+ax^4+bx^3+cx^2+dx+e=0$$ into Bring-Jerrard normal form. Tito Piezas III begins…
Simply Beautiful Art
- 77,026
4
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1 answer
Resultant contains all common roots as linear factors?
Let $f,g \in \mathbb{C}[x,y,z]$ be homogeneous polynomials, so they define projective plane curves $C$ and $D$ in $\mathbb{C}P^2$. We are interested in Bezout's theorem applied to $C \cap D$. Write $f$ and $g$ as polynomials in $z$: $$ f(x,y,z) =…
Herng Yi
- 3,236
4
votes
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
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