# Questions tagged [resultants]

The resultants tag has no usage guidance.

39
questions

-5
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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

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" $...

2
votes

1
answer

126
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### 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

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),\...

1
vote

2
answers

294
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### 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
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### 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

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 ...

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, ...

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 ...

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, ...

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 ...

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 $u\in\mathbb{C}$, $\alpha$ ...

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 ...

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 ...

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 $i\in\{1,2,...,n-1,n\}$, ...

9
votes

2
answers

944
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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

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 ...

4
votes

0
answers

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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

127
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### 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

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 ...

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

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^...

1
vote

0
answers

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### 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

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
$$
...

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 ...

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 ...

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
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0
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181
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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. ...

0
votes

0
answers

554
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### 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 ...

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
...

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 ($\...

8
votes

1
answer

378
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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)=\...

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 ...

2
votes

2
answers

575
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### 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
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### 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 ...

10
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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 ...