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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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Signs in subresultant pseudo-remainder sequence

Subresultant pseudo-remainder sequence is way of computing remainder sequence of two polynomials in $\mathbb{Z}$ and keeping the size of coefficients relatively small, but the signs of the remainders ...
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Resultants to find common root in $\mathbb Z[x_1,\dots,x_n]$

If we have $n$ non-homogeneous polynomials in $\mathbb Z[x_1,\dots,x_n]$ which pairwise have no non-trivial gcd and with a common root in $\mathbb Z^n$ then is it possible to use elimination theory to ...
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The first principal subresultant coefficient of two polynomials

Let $f=f(x), g=g(x) \in \mathbb{C}[x]$, with $\deg(f)=\deg(g)=n \geq 2$. Write $f=(x-a_1)\cdots(x-a_n)$ and $g=(x-b_1)\cdots(x-b_n)$, where $a_i,b_i \in \mathbb{C}$, $1 \leq i \leq n$. Let $\lambda,\...
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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 \{...
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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(...
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Using resultants to show extension of function fields of curves is algebraic.

We are given two irreducible nonsingular plane curves, the zero sets of $f,g\in \bar{k}[x,y]$, with $\bar{k}$ algebraically closed. We have an injective map given on algebras as $\phi^*:C_f\rightarrow ...
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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 ...
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127 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 ...
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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 ...
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29 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 $\tan^{-1}(\frac y x)$?
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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 ...
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1answer
21 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 ...
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253 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 $\mathbb{C}[X,Y]$ ...
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58 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?
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The leading coefficient of a resultant

Let $k$ be a field and $f,g\in k[x,y]$. Let $R(y)=Res_x(f,g)$ be the resultant of $f$ and $g$ considered as polynomials in x over the field $k(y)$. Can you give a nice formula for the leading ...
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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 $\cos\theta$ ...
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73 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-...
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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 ...
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Showing equivalence of resultant between Sylvesterdeterminant and product representation

So, starting with the resultant of 2 polynomials $q(x)$ and $p(x)$ of degree $Q$ and $P$ respectively ($Q<P$) it can be expressed as $$ {\rm Res}_x(q,p) = \prod_{q=1}^{Q} \prod_{p=1}^{P} \left(x_q -...
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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 ...
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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 ...
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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 ...
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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 $...
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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?...
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1answer
53 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 $...
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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 ...
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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, $k \subseteq k(uv) \subseteq k(u)=k(v)$, ...
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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 < ...
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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 ...
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1answer
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Determinant of $xP+yQ$ is resultant of $P$ and $Q$

In my textbook, the resultant $R$ of two polynomials $P$ and $Q$ in $K(X)[Y]$, where $K$ is a field, is defined as the monic generator of the ideal $(P,Q) \cap K[X]$. Is it still true that $ R = \det ...
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198 views

Discriminant and Resultant [closed]

Please help to calculate the discriminant: $$D(\cos(n\arccos(x))$$ How to calculate this according to the discriminant formula through the product of the squares of the root differences? I managed ...
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359 views

What is the determinant of a complex of vector spaces?

I first met the notion of determinants of complexes of vector spaces in the book "Discriminants, Resultants, and Multidimensional Determinants", but I just cannot understand the definition in that ...
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1answer
129 views

Triangular inequality for the Norm in the field extension

Let K(a) be a field extension of field K with algebraic number a. It is well known that Norm(bc)=Norm(b)Norm(c) for b and c in K(a). It can be proved by the property of polynomial resultant. In ...
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Comparing degree of two polynomials via resultant

I am looking for a way to compare the degree of two polynomials via resultant; I.e., given two polynomials $f(x)=f_0+f_1 x^1+\cdots+f_d x^d$ and $g(x)=g_0+g_1x^1+\cdots+g_s x^s$ in $K[f_0,\cdots,f_d,...
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Evaluating an expression given values of symmetric polynomials

Evaluate $\dfrac x{yz} + \dfrac y {xz} + \dfrac z y$ Given, $z+y+x=4, \qquad xyz=-60, \qquad xy+xz+yz=-17$ How do we do this? I found a common denominator, and substituted it for $-60$, but I am ...
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Irrreducibility of resultant

Let $\mathbf{A}$ be a commutative ring with unity, $f(x),g(x)\in\mathbf{A}[x]$. Denote Res($f,g$) be the resultant of $f$ and $g$. Assume that the coefficients of $f$ and $g$ are indeterminates and ...
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Calculating the lead coefficient of the resultant of two polynomials in $\mathbb{Q}[y, l]$

I have two families of polynomials $\{\psi_n \}_{n \in \mathbb{N}}$ and $\{ h_m \}_{m \in \mathbb{N}} $ in $\mathbb{Z}[y, l]$. The polynomials are pretty nice. Each $\psi_n$ is quadratic in $y$, ...
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1answer
148 views

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 $f$...
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216 views

Computation of the resultant through subresultant pseudoremainder sequences algorithm

Let $f,g$ be polynomials of $\mathbb{k}[x]$ where $\mathbb{k}$ is a field. The resultant of $f,g$, $Res(f,g)$, is an element of $\mathbb{k}$ which by definition equals the determinant of the Sylvester ...
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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. ...
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269 views

Resultant of 2 polynomials

I know a little bit about the resultant of two polynomials but I couldn't find any example whatsoever. So I was wondering If you could illustrate the process of finding the resultant of two ...
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1answer
87 views

Resultant of two single-variable polynomials via long division

I need to calculate the resultant of $Q=X^{10}+X^9 + \cdots + 1$ and $P= X^3+X^2+1$ by hand, and I already know it should be $23$. I'm obviously not gonna take the naive way via the coefficient matrix....
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303 views

Discriminant of a product of polynomials

Let $f$ and $g$ be monic polynomials in $\mathbb Z[x]$. Then, \begin{align} \operatorname{disc} (f\cdot g) = \operatorname{disc}(f)\cdot \operatorname{disc}(g) \cdot \prod_i\prod_j(a_i-b_j)^2 , \end{...