Questions tagged [resultants]
The resultants tag has no usage guidance.
39
questions
-5
votes
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answer
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Application of Resultant in Computer Algebra [closed]
Can you guys give me some application of resultant in Computer Algebra, it will be amazing if you guys can give me some paper or book to read more. Thanks so much
2
votes
0
answers
112
views
Resolution of singularities of the resultant locus
We consider projective space of dimension $n$ as the parameter space of degree $n$ polynomials in one variable. Then, I am interested in resolving the singularities of the "resultant locus" $...
2
votes
1
answer
132
views
Library/Database of parametric polynomial systems
Could anyone please recommend a known website where I can find a database/library that has systems of polynomial equations with $n$ variables and $m$ parameters?
I need some real examples to test my ...
6
votes
1
answer
884
views
Resultant of linear combinations of Chebyshev polynomials of the second kind
The Chebyshev polynomial $U_n(x)$ of the second kind is characterized by
$$
U_n(\cos\theta)=\frac{\sin(n+1)\theta}{\sin(\theta)}.
$$
It seems that
$$\operatorname*{Res}_x \left( U_n(x)+tU_{n-1}(x),\...
6
votes
0
answers
736
views
Discriminant of $\alpha P(u) + (z-u) P'(u)$
I'm trying to find a “closed form” of $\textrm{Discriminant}_u(f(u))$, where $f(u) := \alpha P(u) + (z-u) P'(u)$.
Here $P(u)$ is a monic polynomial of degree $d > 1$ with $u\in\mathbb{C}$, $\alpha$ ...
1
vote
2
answers
295
views
A variation on Abhyankar–Moh–Suzuki theorem
The well-known theorem of Abhyankar–Moh–Suzuki says the following:
Let $f=f(t), g=g(t) \in k[t]$, $k$ is a field of characteristic zero.
If $k[f,g]=k[t]$, then $\deg(f) \mid \deg(g)$ or $\deg(g) \mid \...
3
votes
1
answer
308
views
Polynomial function defined recursively by a resultant - is it well defined?
Preliminaries
Let $ n $ be an integer such that $ n \geq3 $. Denote $ \left[ n \right] \equiv \{1,2, \ldots ,n \} $. Let $ P $ be a non-empty subset of $ \left[ n \right] $ such that $ \left|P \right| ...
2
votes
1
answer
95
views
Simple zeroes of complex polynomial $f(\cdot,a)$: condition on $P(a)=\operatorname{Res}_z(f,f')$
A polynomial in the complex variable $z$, whose coefficients are themselves complex polynomials in another complex variable $a$, looks like
$$
f\in\Bbb C[A][Z],\;\;f(z,a)=c_0(a)+c_1(a)z+\cdots+c_n(a)z^...
1
vote
1
answer
244
views
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 \...
0
votes
1
answer
296
views
Dimension of the set of singular hypersurfaces
Let $N$ be the number of degree $d$ monomials in $n$ variables. We can then view each non-zero point in $\mathbb{A}^N_k$ as a degree $d$ homogeneous form, $k$ an algebraically closed field. Let $X$ be ...
1
vote
0
answers
95
views
Expression for the single common root
Let $ \mathbb{F} $ be a field, consider the polynomial ring $ \mathbb{F} \left[ x\right] $ and suppose that the polynomials $f,g \in \mathbb{F} \left[ x\right]$ have degrees $2,2^n$, respectively, ...
2
votes
0
answers
85
views
Methods for multivariate polynomial equations over large 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 ...
2
votes
2
answers
406
views
Resultant of $f(x)$ and $f(-x)$
Let $n \geq 2$ be an integer, and let $f(x) = \prod\limits_{k = 1}^n(x - \alpha_k)$ be a monic irreducible polynomial in $\mathbb Z[x]$, with the property that $f(-\alpha_k) \neq 0$ for any $k = 1, 2, ...
4
votes
2
answers
159
views
Using computer algebra to check if a family of algebras are pair-wise non-isomorphic
Given an infinite field $k$, consider a quiver $\Gamma$ with one vertex and two arrows $x,y$ and define $R=k\Gamma/(x,y)^2.$ This is a three-dimensional $k$-algebra.
Now consider the additive group of ...
1
vote
0
answers
225
views
Polynomial resultants restricted to intervals
The resultant of two polynomials, $R(f,g)$, is a polynomial in the coefficients of $f$ and $g$, and has the property that $R(f,g) = 0$ if and only if $f$ and $g$ share a common root (possibly in an ...
7
votes
1
answer
282
views
Has vol. 3A of Cullis's "Matrices and Determinoids" been scanned and vol. 3B been archived?
This is a borderline question, but I'm going to risk posing it.
Cuthbert Edmund Cullis (1875?-1955?) was a somewhat obscure British mathematician whose opus magnum was a multi-volume treatise called ...
5
votes
1
answer
432
views
Polynomial defined recursively by a resultant
Cross posting from MSE.
Definition:
For any natural number $n\ge 3$, define the polynomial $P_{n}\left(x_1,x_2,...,x_{n-1},x_{n} \right)$, with indeterminates $x_{i}$, where $i\in\{1,2,...,n-1,n\}$, ...
9
votes
2
answers
950
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Polynomials that share at least one root
This is a generalization of an MSE question,
Polynomials that share at least one root.
Let $P(x)$ be a specific polynomial of degree $d$, with given
real coefficients $A_i$ ($A_d=1$), and real roots:
...
1
vote
0
answers
489
views
An explicit formula for characteristic polynomial of matrix tensor product [closed]
Consider two polynomials P and Q and their companion matrices. It seems that char polynomial of tensor product of said matrices would be a polynomial with roots that are all possible pairs product of ...
4
votes
0
answers
104
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Division of bivariate polynomials
The following theorem (lemma 4.2.18 on page 97) is proven in thesis "Computationally efficient Error-Correcting Codes and Holographic Proofs" by Daniel Alan Spielman:
Let $E(X, Y)$ be a polynomial ...
0
votes
0
answers
128
views
Final step in Coppersmith?
In the final step in Coppersmith technique we have $n$ polynomials (possibly non-homogeneous) in $\mathbb Z[x_1,\dots,x_m]$ where $m\leq n$ and using elimination theory we extract the common integer ...
4
votes
1
answer
269
views
Resultants for compactly represented product form polynomials?
Typically computing resultnt of $n+1$ different $n+1$-variate homogeneous polynomials takes $O(poly(\prod_{i=1}^{d_{i}}))$ time where $d_i$ is degree of $i$th polynomial. In certain cases the ...
2
votes
0
answers
484
views
Intersection number of two projective curves using the resultant and tangent lines
For my thesis, I'm working on the intersection of projective plane curves over $\mathbb{C}$. We define the intersection number of projective plane curves (see for example Gibson - Elementary Geometry ...
2
votes
1
answer
91
views
Efficient algorithm to compute resultants of sparse polynomials?
Consider two polynomials $f,g\in\mathbb{F}_2$ of degree $O(2^n)$, with the property that they are extremely sparse (say, only $O(n)$ of the coefficients are non-zero). Is there a way to calculate ...
9
votes
0
answers
298
views
Irreducibility of the Sylvester resultant
If $r$ and $s$ are positive integers, $R$ a commutative ring and $a_0,\dots,a_r$, $b_0,\dots,b_s$ independent variables, we can consider the polynomials $f=\sum_{i=0}^ra_iX^i$ and $g=\sum_{j=0}^sb_iX^...
17
votes
0
answers
220
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GPS calculations under $L^p$ norms
GPS calculations require finding a sphere externally tangent to
four given spheres, an
Apollonian problem
in $\mathbb{R}^3$.
The center of that fifth sphere is one of the $16$ possible solutions to
...
1
vote
0
answers
61
views
Solutions to a certain Birkhoff-interpolation problem
$\newcommand{\CC}{\mathbb{C}}$
Let for $n > 1$ and $m = n-1$
$$
p = x^n + a_1 x^{n-1} + \cdots + a_m x
$$
be a polynomial with $a_i \in \CC$. Call $p^{(i)}(x) = \frac{d^ip}{dx^i}(x)$.
The ...
4
votes
0
answers
99
views
Gröbner bases of resultants and their monomial ideals
$\newcommand{QQ}{\mathbb{Q}}$
Consider the ring $R = \QQ[x, a_1,\ldots,a_m]$ for a certain integer $m$ and the homogeneous polynomial
$$
f = x^{m+1} + \sum_{i=1}^m a_i^i x^{m+1 - i}
$$
Now let
$$
...
17
votes
1
answer
414
views
Splitting the Resultant, as when the Determinant becomes the square of the Pfaffian
The Determinant of an $n\times n$ matrix, viewed as a polynomial in the entries, is irreducible. But when it is restricted to the subspace of alternate matrices, it becomes reducible, actually the ...
12
votes
1
answer
721
views
Determinant is to Pfaffian as resultant is to what?
This is an irresponsible question: I do not have done any thinking on it, or even literature search.
I just became curious whether there is some modification of the notion of a common root of two ...
6
votes
4
answers
663
views
What is the essence of the constant factor in the standard definitions of the discriminant?
Let $f(x) = x^m+\sum_{j=0}^{m-1}f_{m-j}x^j\in P[x]$ be a monic polynomial over a field $P$ and let $f(x) = (x-\alpha_1)\cdot\ldots\cdot(x-\alpha_m)$ be a factorization of $f$ over an extension field $...
7
votes
0
answers
181
views
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. ...
1
vote
3
answers
1k
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Resultant of system with 3 polynomials and 3 variables
Let us say I have a system of 3 polynomials, f1(x,y,z), f2(x,y,z), f3(x,y,z). How to find the resultant of these 3 polynomials? What I mean is: is there any special method to do this? Does the ...
0
votes
0
answers
558
views
How to find solutions for four polynomial equations with four unknown variables using Resultant Theory
Can I use resultant theory (or polynomial resultant method) to find solutions for four simultaneous polynomial equations with four unknown variables?
So far, I could only find examples which uses two ...
2
votes
2
answers
577
views
The resultant of two degree n and n - 1 functions in two variables of t
I'm currently studying the implicitization of bezier curves (that is, finding a function that f(x, y) = 0 for any x and y pairs of a curve p(t)) as part of an algorithm for curve intersection. The ...
3
votes
2
answers
378
views
When is the Wendt binomial circulant determinant divisible by 3?
The Wendt binomial circulant determinant $W_n$ can be defined quite simply as a resultant:
$$ W_n = \operatorname{res}(x^n-1, (x+1)^n-1). $$
Truer to its name, one may also define it as the ...
1
vote
1
answer
364
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Irreducibility of a resultant of real and imaginary parts of a characteristic polynomial
The following question is motivated by the study of a stability border for a robust linear time-invariant control system.
Let us we have an affine family of $n\times n$ matrices with indeterminate ($\...
10
votes
5
answers
3k
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Multipolynomial resultants
We know that the resultant of two polynomials can be computed as the determinant of their Sylvester matrix ( http://en.wikipedia.org/wiki/Sylvester_matrix ). How do we compute the resultant of more ...
8
votes
1
answer
381
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Combinatorics of resultants
This is a crosspost of https://math.stackexchange.com/questions/446470/combinatorics-of-resultants which received no answer. [EDIT: I deleted the initial copy of the question on MathSE].
Let $f(z)=\...