Questions tagged [resultants]

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2 votes
1 answer
84 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
752 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),\...
  • 437
1 vote
2 answers
269 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 \...
  • 2,603
3 votes
1 answer
283 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
92 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^...
  • 751
1 vote
1 answer
237 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 \...
  • 2,603
0 votes
1 answer
198 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 ...
  • 3,159
1 vote
0 answers
84 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
64 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 ...
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2 votes
2 answers
378 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, ...
  • 1,483
4 votes
2 answers
144 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 ...
6 votes
0 answers
702 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
0 answers
219 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 ...
  • 51
7 votes
1 answer
274 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
394 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
886 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: ...
1 vote
0 answers
342 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 ...
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4 votes
0 answers
84 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 ...
0 votes
0 answers
124 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 ...
  • 13.1k
4 votes
1 answer
259 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 ...
  • 13.1k
2 votes
0 answers
390 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 ...
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2 votes
1 answer
80 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 ...
  • 131
9 votes
0 answers
265 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^...
1 vote
0 answers
57 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
89 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
366 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 ...
  • 48.5k
12 votes
1 answer
694 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
652 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
164 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. ...
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0 votes
0 answers
526 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 ...
  • 1
17 votes
0 answers
216 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 ...
1 vote
1 answer
350 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 ($\...
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8 votes
1 answer
364 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)=\...
  • 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 ...
  • 21
2 votes
2 answers
556 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
374 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 ...
  • 4,567
10 votes
5 answers
2k 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 ...
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