Questions tagged [resultant]

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

Filter by
Sorted by
Tagged with
0
votes
1answer
29 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) $k[f',g']=k[t]$...
0
votes
0answers
16 views

$D(A)=(-1)^{\frac{n(n-1)}{2}}a_{n}^{-1}R(A,A^{'})$

Suppose $A(x)=a_{n}x^{n}+a_{n-1}x^{n-1}+...+a_{0}=\prod_{i=1}^{i=n}(x-x_{i})$. I need to prove that $D(A)=(-1)^{\frac{n(n-1)}{2}}a_{n}^{-1}R(A,A^{'})$, where $D(A):=a_{n}^{2n-2}\prod_{i<j}(x_{i}-x_{...
1
vote
1answer
56 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 ...
0
votes
0answers
21 views

Coprime polynomials of consecutive degrees

Let $f,g \in k[t]$ be two polynomials, $k$ a field of characteristic zero. Assume that $f$ and $g$ satisfy the following two conditions: (i) $f$ and $g$ are coprime, namely, they have no common root (...
1
vote
1answer
36 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 ...
1
vote
1answer
78 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)=...
1
vote
0answers
14 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 ...
0
votes
0answers
21 views

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 $g(x)$ ...
1
vote
1answer
65 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 \...
2
votes
1answer
104 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 ...
0
votes
0answers
22 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 ...
0
votes
1answer
48 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 ...
0
votes
0answers
30 views

Discriminant of $X=Z(f)$ in terms of join, intersection and projection.

Say we have $X\subset \mathbb{CP}^{n}$ a projective variety, we may even assume it is a hypersurface for convenience. Let $\pi:\mathbb{CP}^{n}\to\mathbb{CP}^{n-1}$ be the standard projection. I am ...
0
votes
1answer
81 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 ...
1
vote
0answers
22 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 ...
1
vote
1answer
190 views

Matrices with eigenvalue of multiplicity $k$ are algebraic subset of special linear group.

In the previous exercise, I have showed that the special linear group $SL_n$ is a closed subvariety of $Mat(n,K)$ where $K$ is an algebraically closed field with characteristic zero. Now, I have to ...
0
votes
0answers
29 views

Finding $a,b \in k$ such that $\gcd(f(t)-a,g(t)-b)=t$

Let $f=f(t),g=g(t) \in k[t]$, whrer $k$ is a field of characteristic zero. Assume that: (i) $k(f,g)=k(t)$. Therefore, there exist $P(X,Y),Q(X,Y) \in k[X,Y]$, such that $t=\frac{P(f,g)}{Q(f,g))}$. (...
0
votes
0answers
7 views

If $k(f(t),g(t))=k(t)$, then what can be said about $Q(X,Y)$ such that $t=\frac{P(f,g)}{Q(f,g)}$?

Let $f=f(t), g=g(t) \in k[t]$, where $k$ is a field of characteristic zero. Assume that $k(f(t),g(t))=k(t)$. Therefore, there exist $P(X,Y), Q(X,Y) \in k[X,Y]$ with $Q(X,Y) \neq 0$ such that $t=\...
1
vote
0answers
40 views

Resultant of multivariate polynomials

I know little about resultant. As for as I know, if two univariate polynomials $f,g$ have a common zero iff $Res(f,g)=0$, is it right? Now I see an example using resultant of two multivariate ...
2
votes
1answer
48 views

When $\mathbb{C}(f(t),g(t))=\mathbb{C}(t)$ imply that there exist $a,b \in \mathbb{C}$ such that $\deg(\gcd(f(t)-a,g(t)-b))=2$?

Assume that $f=f(t),g=g(t) \in \mathbb{C}[t]$ satisfy the following two conditions: (1) $\deg(f) \geq 2$ and $\deg(g) \geq 2$. (2) $\mathbb{C}(f,g)=\mathbb{C}(t)$. After asking this and then this ...
0
votes
0answers
47 views

Concerning possible degrees of certain $f,g \in k[t]$

Let $k$ be a field of characteristic zero, $k \in \{\mathbb{R},\mathbb{C}\}$. Ler $f=f(t),g=g(t) \in k[t]$ such that: (1) The ideal in $k[t]$ generated by $f',g'$ (formal derivatives of $f,g$, ...
1
vote
0answers
105 views

Multivariate Polynomial Defined Recursively By A Resultant Function

Suppose that the characteristic of the field $\mathbb{F}$ is not $2$. Definition: For any natural number $n$, such that $3 \le n$, and for any $x_{i} \in \mathbb{F}$, where $i\in\{1,2,...,n-1,n\} $, ...
-1
votes
1answer
16 views

Misconception related to Resultant of Vectors [closed]

I have read in many books that the triangle law gives us resulatant of two vectors. $Question$ : How do we knew that the resultant is sum of two vectors not the product of the two vectors?? Clearly ,...
0
votes
0answers
37 views

Computing subresultants in WolframAlpha

Let $f(x)=(x-a_1)(x-a_2)\cdots(x-a_m)$ and $g(x)=(x-a_{m+1})(x-a_{m+2})\cdots(x-a_n)$, where $m,n \in \mathbb{N}$, $m < n$, $a_i \in F$, $F$ is a field of characteristic zero (for example, $k \in \{...
4
votes
0answers
97 views

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(...
0
votes
0answers
37 views

Resultant of two polynomials - generating matrix

Calculate the resultant of x^2+1, x^2+3x I know the resultant is the determinant of the matrix below but don't understand how that matrix is generated. Could someone please explain? 1 0 0 0 0 1 3 0 ...
4
votes
2answers
183 views

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 ...
0
votes
0answers
78 views

Root of resultant if and only if common root

I have solved (maybe) this exercise, but I'm not sure about my proof. The exercise text is: Let $K$ be a field and let $f,g\in K[x_1,\dots ,x_n]$ with $x_n$ actually appearing in $f$ and $g$. Show ...
1
vote
1answer
30 views

Calculating Sums Of Maths [closed]

Slightly confused. Would I be correct in saying to find the direction angle of the resultant, I would need to do $\tan^{-1}(\frac y x)$?
1
vote
1answer
42 views

Calculating Sums Of Maths

I need to use Pythagoras’ theorem to calculate the magnitude of the resultant. I've calculated the horizontal and vertical components. I've attached the image here: I've added up by vertical ...
0
votes
1answer
29 views

Find forces with given their resultant

the resultant of two force P and 10 KN is 15KN inclined 30 degree to the 10KN force find the magnitude and detection of force i draw my sketch and try to use cosine rule but i can't find please ...
5
votes
1answer
385 views

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 $\mathbb{C}[X,Y]$ ...
4
votes
2answers
103 views

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?
0
votes
1answer
563 views

How do we get a cos component and a sin component when we resolve a vector?

So, I'm trying to understand this. In the following we've an image, with a vector $v_0$ and we can see the components of the vectors have trigonometric ratios such as $\sin\theta$ and $\cos\theta$ ...
2
votes
0answers
82 views

Common zeros of two univariate polynomials over $\mathbb{C}(y)$

Let $f=f(x), g=g(x) \in \mathbb{C}[y][x] \subset \mathbb{C}(y)[x]$, where $\{x,y\}$ are variables over $\mathbb{C}$, and each of $\{f,g\}$ is of $x$-degree $\geq 1$, namely, $f$ and $g$ are non-...
4
votes
1answer
211 views

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 ...
1
vote
1answer
41 views

Computing the resultant of two polynomials of degree $4$

Let $f=f(t)=\sum_{i=0}^{4}a_ix^i ,g=g(t)=\sum_{j=0}^{4}b_jx^j \in L[x]$, $L$ is a commutative ring. I would like to compute the resultant of $f$ and $g$. I wish to check a few examples in low ...
3
votes
0answers
118 views

A relation between common roots of $f$ and $g$ and common roots of $f-\lambda$ and $g-\lambda$

Let $f=f(t),g=g(t) \in k[t]$, $f \neq g$, and $\lambda,\mu \in k$. (1) Is there a relation between $\gcd(f,g)$ and $\gcd(f-\lambda,g-\mu)$? If we only wish to know what is the relation between the ...
3
votes
2answers
214 views

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 ...
3
votes
1answer
152 views

Characterizing $f$ and $g$ such that $\deg(\gcd(f,g)) \geq 2$.

Let $f=f(t),g=g(t)\in \mathbb{C}[t]$, with $\deg(f),\deg(g) \geq 3$. A known result about the resultant of $f$ and $g$ says the following: The resultant of $f$ and $g$ is $0$ if and only if $f$ and $...
2
votes
0answers
133 views

Multivariable Resultant

I want to compute the resultant of $f_1,\cdots, f_m \in K[x_1,\cdots x_n]$ where $f_1, \cdots,f_m$ are forms of degrees $d_1,\cdots ,d_n$ respectively. Does anyone have any idea on how could I do this?...
1
vote
1answer
67 views

Writing the discriminant of an integer cubic polynomial with no double root as a combination of polynomials

Let $f(X)=X^3+aX^2+bX+c \in \mathbb Z[X]$ be a polynomial such that $f(X)$ and $f'(X)$ has no common root in $\mathbb C$. Let $\alpha_i$ , $i=1,2,3$ are the distinct roots of $f$ in $\mathbb C$. Let $...
0
votes
0answers
69 views

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 ...
0
votes
0answers
73 views

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, ...
2
votes
0answers
192 views

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 ...
1
vote
0answers
100 views

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 ...
2
votes
1answer
373 views

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 ...
0
votes
0answers
32 views

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, $k \subseteq k(uv) \subseteq k(u)=k(v)$, ...
0
votes
1answer
82 views

Do two polynomials $A$ and $B$ share a root when $AP+BQ=0$ for some polynomials $(P,Q) \neq (0, 0)$ with $\deg P < \deg B$, $\deg Q < \deg A$?

My question Let $A$ and $B$ be polynomials of degrees $d$ and $e$ respectively. If there exists a pair of polynomials $(P, Q) \neq (0, 0)$ such that $AP+BQ = 0$, with $\deg P < e$ and $\deg Q < ...
1
vote
0answers
111 views

Reduce the size of resultant with polynomial of two variables.

I have a problem in which I have to calculate given two polynomials $f,g \in \mathbb{Q}[X]$ a polynomial with roots the product of the roots of $f$ and $g$. My reasoning shows that this reduces to ...