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Questions tagged [resultant]

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

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When does the eliminant (resultant) from three variables into two variables vanish?

Definition. Consider a ring $\mathcal{R}$ and polynomials $P,Q\in \mathcal{R}[x]$. We define the eliminant $\mathrm{Elm}(P,Q)$ of $P$ and $Q$ by the determinant of their Sylvester matrix. If $P(x)=\...
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Discriminant of quadratic polynomial through the resultant of (f,f')

I would like to calculate the discriminant of $f := ax^2+bx+c$ using the resultant of $(f, f')$. The formula I found for this is $a_n^{-1}(-1)^{\frac{n(n-1)}{2}} \text{res}_x(f(x), f'(x))$. $\text{res}...
lkksn's user avatar
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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 ...
Stéphane Laurent's user avatar
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1 answer
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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 ...
BD107's user avatar
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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 ...
Valere's user avatar
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A lower bound of the distance between 2 distinct roots of 2 distinct polynomials

Here is one of my homework in the computer algebra class: Let $f(x),g(x)\in \mathbb{Z}[x]$ be of positive degree $m,n$, and $f(\alpha)=g(\beta)=0$, where $\alpha\neq \beta$ are real. Show that $$ |\...
Zoudelong's user avatar
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Resultant bounded by norms

Prove for $f,g \in \mathbb{Z}[x], \deg f = n, \deg g = m$: $$|res(f,g)| \leq ||f||_2^m ||g||_2^n \leq (n+1)^{m/2} (m+1)^{n/2} |f||_\infty^m ||g||_\infty^n$$. Now I have used as definition for the ...
Magne Seier's user avatar
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Prove inequality about degrees of polynomials and resultant

Let $f,g \in F[x,y], \deg_xf=n,\deg_xg=m,d\in \mathbb{N}:\deg_y f, \deg_y g \leq d$. Then $\deg_y res_x(f,g) \leq (n+m)d$. I have $fg=\sum_{i=0}^n \sum_{j=0}^m f_{ij} g_{ij} x^i y^j$ (Is this a ...
Magne Seier's user avatar
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Prove identity involving resultant of polynomials

Consider polynomials $f,g \in k[x]$ of positive degrees $m,n$ respectively. Let $I$ denote the ideal in $k[x]$ generated by $f$, and let $\mu$ denote the multiplication map $$\mu:k[x]/I \to k[x]/I,h+I ...
mathtronaut's user avatar
4 votes
1 answer
164 views

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\...
Arrow's user avatar
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resultant of two co-prime polynomials over $\mathbb{Z}[x]$ , is there anything special about this resultant?

let $f(x)=x^3-1$ and $g(x)=a_0+a_1x+a_2x^2$, where $f(x),g(x) \in \mathbb{Z}[x]$, and $\textrm{GCD}(f(x),g(x))=1$. From all this, is it possible to infer anything about the resultant $\textrm{Res}(f(x)...
eagle I 's user avatar
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Explicit bivariate resultant example

I have two polynomials modulo a prime $p$ with a common root $(x_0,y_0)$ with $|x_0|,|y_0|<\sqrt{p}$ and there are no other roots in $\mathbb F_p$ or no other roots of this size in $\mathbb F_p$ (...
Turbo's user avatar
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$(f,g)$ is unimodular with $f$ unitary then $R(f,g)\notin \mathfrak{m}$

Let $A$ be a commutative ring with unity and $(f,g)\in A[X]^2$ is unimodular with $f$ unitary then the resultant of $(f,g)$ is not in the maximal ideal $\mathfrak{m}$ of $A$, i.e., $R(f,g)\notin \...
Lalbahadur Sahu's user avatar
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1 answer
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Statement on resultants: for polynomials A, B, res(A, B) belongs to the ideal generated by A and B

Consider the following statement on resultants, given for example on the Wikipedia page for resultants Let $A, B$ be polynomials in $R[X]$ where $R$ is a commutative ring. Let $A, B$ have degrees $d, ...
Mr Lolo's user avatar
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finding the resultant using paralellogram method to find the resultant

for some reason I don't understand my prof again... we were doing online class since there is a super typhoon so they suspended face to face, since it is not face to face it is hard to understand what ...
ciumai's user avatar
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2 votes
2 answers
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Resultants - if $a,b$ have degrees $m,n$, why are there polynomials $u,v$ of degrees $m-1,n-1$ so that $Res(a,b)=av-bu$?

Let $R$ be a commutative ring. Suppose I have polynomials $a\left(x\right),b\left(x\right) \in R\left[x\right]$ of degrees $m,n$, respectively. My reading states that there then exist polynomials $u\...
Aaron Russell's user avatar
1 vote
2 answers
106 views

multiplicity of the intersections between $p=x_0x_1^2+x_1x_2^2+x_2x_0^2\,\,$ and $\,\,q=-8(x_0^3+x_1^3+x_2^3)+24x_0x_1x_2$ in $\mathbb{P^2(K)}$

in $\mathbb{P^2(K)}$ where $\mathbb{K}$ is an algebraically closed field and $[x_0,x_1,x_2]$ the homogeneous coordinates, consider the following (homogeneous) polynomials: $p=x_0x_1^2+x_1x_2^2+x_2x_0^...
AleVanDerBauch's user avatar
1 vote
0 answers
64 views

$\mathrm{Res}(f, g, h)$ as a product of three resultants?

I'm currently working with resultants, which I define as follows: let $k$ be a field, $V$ be a 2-dimensional vector space over $k$, and let $S^dV^*$ be the $d$-th symmetric power of dual space, i.e. ...
Aleksei Kubanov's user avatar
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Finding the kernel of a given map

Let $\mathbb K$ be an algebraic closed field and let $f,g\in\mathbb K[X,Y]$ be two polynomials so that $V_{\mathbb K}(f)$ and $V_{\mathbb K}(g)$ don't have common irreductible components and $(0,0)\in ...
Diego Trujillo's user avatar
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1 answer
192 views

Show that the sum of 2 algebraic integers is an algebraic integer using resultants

I've started working on algebraic numbers very recently for a memoir, that is I didn't study them in class. I need them, and particularly algebraic integers, to prove a couple propositions which aren'...
Rhaena's user avatar
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2 answers
115 views

$x^4+y^4+z^4=\frac{m}{n}$, find $m+n$.

$x^4+y^4+z^4$=$m\over n$ x, y, z are all real numbers, satisfying $xy+yz+zx=1$ and $5\left(x+\frac{1}{x}\right)=12\left(y+\frac{1}{y}\right)=13\left(z+\frac{1}{z}\right)$ m, n are positive integers ...
ishandutta2007's user avatar
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1 answer
81 views

$\mathbb{C}(f,g)=\mathbb{C}(t)$ and $(f'(t),g'(t)) \neq 0$, but $\mathbb{C}[f,g]\subsetneq \mathbb{C}[t]$

Actually, the following question is a consequence of this MO question. Recall the following result: $\mathbb{C}[f(t),g(t)]=\mathbb{C}[t]$ iff $(f'(t),g'(t))\neq 0$ and $t\mapsto (f(t),g(t))$ is ...
user237522's user avatar
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5 votes
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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.
Cel's user avatar
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A variation on $k(x^2,x^3)=k(x)$

Let $k$ be a field of characteristic zero, for example $k=\mathbb{R}$ or $k=\mathbb{C}$. Of course, $k(x^2,x^3)=k(x)$, since $x=\frac{x^3}{x^2}$. Let $f_1,\ldots,f_n,g_1,\ldots,g_m \in k[x]$, $n,m \...
user237522's user avatar
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0 votes
1 answer
40 views

I'm looking to begin making product spaces of polynomials..

I'm studying a definition of a resultant so I have to compute the product of two vector spaces $\mathcal{P}^{i}$ with elements that are polynomials of degree less than $i$. If $1+x \in \mathcal{P}^{2}$...
fia presheaf's user avatar
0 votes
1 answer
75 views

Three vectors resultant force

Im trying to solve this question. My question is: should i relocate vector (5N) from third quadrant to first quadrant? Because it is given with its head touching the axis, if i keep it in third ...
John's user avatar
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2 votes
1 answer
640 views

How is the resultant obtained from a pair of quadratic polynomials?

I'm reading about resultant theory from Peter Stiller's notes (Link to pdf). The author mentions that, given two univariate polynomials $$f(x)=a_r x^r+ ...+a_1 x+ a_0, \qquad g(x)=b_s x^s+ ...+b_1 x+ ...
glS's user avatar
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1 vote
1 answer
69 views

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 $R(f,g)$ ...
DBXz's user avatar
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1 vote
1 answer
533 views

Projection map sends closed sets to closed sets

I am reading Algebraic Geometry, a first course by Joe Harris. In the section on projections, he talks about an application of elimination theory to prove that image of a projection map $\pi: Y \times ...
Geet Thakur's user avatar
1 vote
0 answers
110 views

The best methods for multivariate polynomial equations over finite fields

I am trying to get a rough overview of the best methods one might use to find solutions of multivariate polynomial equations over large finite fields. We can suppose for simplicity that the given ...
sugyman's user avatar
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1 vote
1 answer
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Prove that $\operatorname{res}(A(ax),B(ax)) = a^{de}\operatorname{res}(A(x),B(x))$, using the fact that $\sum(i-\sigma(i))=0$

I have been asked to prove the following result: $$\operatorname{res}(A(ax),B(ax)) = a^{de}\operatorname{res}(A(x),B(x))$$ where $a,e$ are the degrees of the polynomials $A(x)$ and $B(x)$, using the ...
ben huni's user avatar
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2 votes
2 answers
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How can I show two polynomials common roots with a given matix and resultant?

How can I show, that if $u:= ax + b$ and $v:= x^2 + cx + d$ , $(a, b, c, d ∈ \mathbb{R})$ polinoms have common root, if and only if $det(A[u, v]) = 0$, where $A[u, v] = \begin{bmatrix} a & b &...
Zaevras's user avatar
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1 answer
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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 ...
Manar's user avatar
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4 votes
0 answers
175 views

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 ...
Aaron Cao's user avatar
4 votes
0 answers
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 ...
Frenkel Péter's user avatar
0 votes
1 answer
47 views

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 ...
W-t-P's user avatar
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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?
Diplo's user avatar
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6 votes
1 answer
273 views

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 ...
Randy Marsh's user avatar
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familiy of polynomials separable

Let $0<m<n$ in $\mathbb{N}$ and $a_i,b_j\in\mathbb{C}$ for $0\leq i\leq m$ and $0\leq j\leq n$ and $a_m,b_n\neq 0$. Consider $(\sum_{i=0}^{m}a_i (X+Y)^i)(\sum_{j=0}^{n}b_i (1+Y)^i)(\sum_{j=0}^{n}...
rike's user avatar
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0 votes
1 answer
43 views

$f,g \in k[t]$ 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) $k[f',g']=k[t]$...
user237522's user avatar
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1 vote
1 answer
631 views

Where Can I Find Van der Waerden's Modern Algebra, Volume I, Chapter 5 Online?

I am looking for an electronic copy of Van der Waerden's Modern Algebra, Volume 1, Chapter 5. Due to the pandemic, I can't get it from a library. I am not an expert on copyright law, but my ...
Tri's user avatar
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1 vote
1 answer
50 views

$f,g \in k[t]$ with $k(f,g)=k(t)$, $\deg(f)=2$ and $\deg(g)=3$

Let $f=f(t),g=g(t) \in k[t]$, $k$ is a field of characteristic zero. Assume that the following two conditions are satisfied: (i) $\deg(f)=2$ and $\deg(g)=3$. (ii) $k(f,g)=k(t)$. Question: Is it ...
user237522's user avatar
  • 6,595
1 vote
1 answer
197 views

$f,g \in k[t]$ such that $\deg(f)=\deg(g)$ and $k(f,gt)=k(t)$

Let $f=f(t), g=g(t) \in k[t]$ be two nonzero polynomials over a field $k$ of characteristic zero. Assume that the following two conditions are satisfied: (i) $\deg(f)=\deg(g) \geq 1$. (ii) $k(f,gt)=...
user237522's user avatar
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1 vote
0 answers
27 views

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 ...
Descartes Before the Horse's user avatar
2 votes
1 answer
414 views

Understanding resultant

Let us denote by $P_k$ the vector space polynomials over $\mathbb{K}$ of the degree $< k$. Take $f, g \in \mathbb{K}[x]$, $deg(f) = n,$ $deg(g) = m$ and define the linear map $$T : P_m \...
Invincible's user avatar
  • 2,642
2 votes
1 answer
128 views

Find $F(x,y)$ such that $F(p(t),q(t))=0$

$p(x),q(x)$ be real polynomials with degree $\leq n (n> 1)$, both are non constants. How to find a $F(x,y)\neq 0$ such that $F(p(t),q(t))=0, \forall\ t\in\Bbb R$, where $F\neq 0$ is real polynomial ...
xldd's user avatar
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0 answers
37 views

Algebraic degree of unique solution of system of linear equations with algebraic coefficients

Let $\varphi_1,\ldots,\varphi_n$ be the roots of a monic polynomial $p$ of degree $n$ with integer coefficients. $\varphi_i$ are thus, by definition, algebraic integers. Let $a_i$, $i=1,\ldots,n$, be ...
Eckhard's user avatar
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0 votes
1 answer
125 views

resultant invariance $Res(f+gh,g) = Res(f,g)$ [closed]

I recently started learning the applications of the resultant and I found the following statement: Let $f(x),g(x) \in \mathbb{K}[x]$ (where $\mathbb{K}$ is some field) and $deg(f) \geq deg(g)$ prove ...
ned grekerzberg's user avatar
0 votes
1 answer
316 views

How one vector is related to two vectors?

Imagine that I have a vector $\bf{v}$ which is always between two vectors $\bf{v}_1$ and $\bf{v_2}$. I would like to know how vector $\bf{v}$ can be obtained from $\bf{v_1}$ and $\bf{v_2}$. If we ...
Joe Hofstrand's user avatar
1 vote
0 answers
28 views

Translating Operations on Solutions to that of Equations

Let $a, b$ be two algebraic numbers, $f(x), g(x)$ their minimal polynomials over $\mathbb{Q}$ respectively. Suppose the set of roots of $f$ is $$A = \{a=a_1, a_2, \cdots, a_n\}$$ and the set of roots ...
Student's user avatar
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