Questions tagged [resultants]

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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
Luật Trần Văn's user avatar
2 votes
0 answers
109 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" $...
Asvin's user avatar
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2 votes
1 answer
126 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 ...
Ayoola Jinadu's user avatar
6 votes
1 answer
875 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),\...
W. Wang's user avatar
  • 437
1 vote
2 answers
294 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 \...
user237522's user avatar
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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| ...
PalmTopTigerMO's user avatar
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^...
Joe's user avatar
  • 759
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 \...
user237522's user avatar
  • 2,735
0 votes
1 answer
291 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 ...
Johnny T.'s user avatar
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1 vote
0 answers
94 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, ...
PalmTopTigerMO's user avatar
2 votes
0 answers
82 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 ...
sugyman's user avatar
  • 21
2 votes
2 answers
403 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, ...
Anton's user avatar
  • 1,553
4 votes
2 answers
155 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 ...
Sergey Guminov's user avatar
6 votes
0 answers
734 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 $u\in\mathbb{C}$, $\alpha$ ...
Fll'Yissetat's user avatar
1 vote
0 answers
224 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 ...
opti's user avatar
  • 51
7 votes
1 answer
279 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 ...
darij grinberg's user avatar
5 votes
1 answer
428 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 $i\in\{1,2,...,n-1,n\}$, ...
PalmTopTigerMO's user avatar
9 votes
2 answers
944 views

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: ...
Joseph O'Rourke's user avatar
1 vote
0 answers
466 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 ...
Andrew S.'s user avatar
  • 119
4 votes
0 answers
101 views

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 ...
Maxim Nikitin's user avatar
0 votes
0 answers
127 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 ...
Turbo's user avatar
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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 ...
Turbo's user avatar
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2 votes
0 answers
479 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 ...
Onnovz's user avatar
  • 21
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 ...
Daniel's user avatar
  • 131
9 votes
0 answers
292 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^...
Mariano Suárez-Álvarez's user avatar
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 ...
Jürgen Böhm's user avatar
4 votes
0 answers
98 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 $$ ...
Jürgen Böhm's user avatar
17 votes
1 answer
409 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 ...
Denis Serre's user avatar
  • 51.3k
12 votes
1 answer
715 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 ...
მამუკა ჯიბლაძე's user avatar
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 $...
Mikhail Goltvanitsa's user avatar
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. ...
xyz's user avatar
  • 306
0 votes
0 answers
554 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 ...
Joy's user avatar
  • 1
17 votes
0 answers
220 views

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 ...
Joseph O'Rourke's user avatar
1 vote
1 answer
362 views

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 ($\...
probably's user avatar
  • 403
8 votes
1 answer
378 views

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)=\...
minar's user avatar
  • 492
1 vote
3 answers
1k views

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 ...
Sha's user avatar
  • 21
2 votes
2 answers
575 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 ...
Lucas McCarthy's user avatar
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 ...
aorq's user avatar
  • 4,924
10 votes
5 answers
3k views

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 ...
Andrew's user avatar
  • 103