12
votes
Accepted
Resultant of linear combinations of Chebyshev polynomials of the second kind
Since $U_n + t U_{n-1}$ is of degree $n$ and $\sum_{k=0}^{n-1} U_k$ is of degree $n-1$ with leading coefficient $2^{n-1}$, the resultant factors as
$$ 2^{n(n-1)} (-1)^{n(n-1)} \prod_{j=1}^{n-1} (U_n(...
- 114k
12
votes
Accepted
Resultant of $f(x)$ and $f(-x)$
We have $\mathrm{Res}(f(x),f(-x))=2^n a_n P(\alpha)^2$, where
$P(\alpha)=\prod_{1\leq i<j\leq n}(\alpha_i+\alpha_j)$. By
e.g. the case $d=2$ of Exercise 7.30 in Enumerative Combinatorics,
vol. 2, ...
- 50.8k
12
votes
Splitting the Resultant, as when the Determinant becomes the square of the Pfaffian
I think it is better to homogenize and view $P$, $Q$ as binary forms in $\mathbb{C}[X_1,X_2]$. An invertible $2\times 2$ matrix $g$ acts on the variables $x_i$ by $(gx)_i=\sum_{j=1,2}g_{ij}x_j$ and on ...
11
votes
Accepted
Determinant is to Pfaffian as resultant is to what?
Pfaffian resultant formulas are obtained in Resultants and Chow forms via Exterior Syzygies (2001), where the polynomials are represented by coordinates on a Grassmanian manifold.
- 188k
11
votes
Accepted
What is the essence of the constant factor in the standard definitions of the discriminant?
As Robert said, if you want everything to work in $\mathbb Z[f_0,\ldots,f_m]$, you need that factor. I'll also mention that your polynomial indexing is messed up, you probably meant the sum to go from ...
- 47.4k
10
votes
What is the essence of the constant factor in the standard definitions of the discriminant?
The factor $f_0^{2m-2}$ makes the discriminant a polynomial in the coefficients $f_0, \ldots, f_m$.
- 54.2k
9
votes
Accepted
Polynomials that share at least one root
The locus of real polynomials $p(x)$ sharing a root with $P(x)$ is the union of the hyperplanes $H_\alpha : p(\alpha)=0$, where $\alpha$ runs over the roots of $P(x)$. This is an arrangement of ...
- 20.8k
6
votes
Accepted
Has vol. 3A of Cullis's "Matrices and Determinoids" been scanned and vol. 3B been archived?
Q1: Volume 3 part 1 (a.k.a. volume 3A) has been digitized and reissued as a paperback by Cambridge UP, see Amazon. The digital version is online in the HathiTrust digital library, but with limited "...
- 188k
4
votes
What is the essence of the constant factor in the standard definitions of the discriminant?
Perhaps you will find the following helpful, which expands a bit on Robert Israel's answer: Write $f(x)=f_0x^m+\ldots+f_m=f_0\cdot(x-\alpha_1)\cdots(x-\alpha_m).$ Multiplying out the RHS we get $f(x)...
- 868
3
votes
Accepted
Defining polynomial of compositum of splitting fields
For monic polynomials $f,g$ define $f*g=\mathrm{Res}_y(f(y),g(x-y))$. This is an associative and commutative operation on univariate monic polynomials with neutral element $x$. If $f=\prod_{i=1}^m(x-\...
- 651
3
votes
Accepted
Library/Database of parametric polynomial systems
There is a database of polynomial systems, which comes with PHCpack by Jan Verschelde.
- 34.3k
3
votes
Accepted
Simple zeroes of complex polynomial $f(\cdot,a)$: condition on $P(a)=\operatorname{Res}_z(f,f')$
No, it's not true, as is shown by the polynomial $f(z,a):=(z-a)(z-1)^2$.
- 5,428
3
votes
Accepted
A variation on Abhyankar–Moh–Suzuki theorem
The answer is no. You can simply choose $$f=t(t^2-t+1)$$
$$g=t(t^2+1)$$
These satisfy all your conditions, but $\mathbb{C}[f,g]\subsetneq \mathbb{C}[t]$.
Let us check this:
$(1)-(2)$: $f=tf_1$ and $g=...
- 7,722
3
votes
Polynomial defined recursively by a resultant
This question is close to a central one in Rational Trigonometry, which is to determine the relations between successive quadrances between a cyclic list of points on a line. Here quadrance refers to ...
2
votes
What is the essence of the constant factor in the standard definitions of the discriminant?
A trivial observation, which was not pointed out so far: the non-normalized discriminant is 'better' in that it comes 'closer' to being a 'homomorphism' $P[x]\rightarrow P$, though it 'still' is not ...
- 6,051
2
votes
Polynomials that share at least one root
I tried to make a 3D image for
$P(x)=x^3+3 x^2-2 x-1$. The set consists of three planes,
each tangent to the discriminant surface.
But it became too visually complex, partly because
the discriminant
...
2
votes
Using computer algebra to check if a family of algebras are pair-wise non-isomorphic
For this and other Macaulay2-related questions, I highly recommend the Macaulay2 google group.
In general, there are some "exploratory" techniques (i.e. not quite a proof) which are more ...
- 396
2
votes
Accepted
Polynomial function defined recursively by a resultant - is it well defined?
Notice that
$$
\let\eps\varepsilon
q(x_1,x_2,x_3)
=-\bigl(\sqrt{x_1}+\sqrt{x_2}+\sqrt{x_3}\bigr) \bigl(-\sqrt{x_1}+\sqrt{x_2}+\sqrt{x_3}\bigr) \bigl(\sqrt{x_1}-\sqrt{x_2}+\sqrt{x_3}\bigr) \bigl(-...
- 23.7k
2
votes
Accepted
Dimension of the set of singular hypersurfaces
Given a smooth projective variety $X\subset\mathbb{P}^{N-1}$, let $I(X)\subset X\times(\mathbb{P}^{N-1})^*$ denote the locus of pairs $(x,H)$ where $x\in H$ and $T_x(X)\subset T_x(H)$; here we are ...
- 1,566
1
vote
Accepted
Conditions for resultants of bivariate quadratics to be perfect squares
At very least it is necessary that each of the following three numbers is a square:
$$a_{02}^2 b_{20}^2-a_{02} a_{11} b_{11} b_{20}-2 a_{02} a_{20} b_{02} b_{20}+a_{02} a_{20} b_{11}^2+a_{11}^2 b_{02} ...
- 34.3k
1
vote
Application of Resultant in Computer Algebra
Perhaps this is not the best forum to ask such a question; nevertheless I can provide some answer. You may like the paper Computing resultant matrices with Macaulay2 and Maple by Busé, which not only ...
- 2,992
1
vote
A variation on Abhyankar–Moh–Suzuki theorem
Since you (OP) seem to be interested in Lüroth’s theorem, you might want to look at a "constructive" proof that I just wrote up from Schinzel's Selected topics on polynomials. It seems using ...
- 5,339
1
vote
Accepted
A variation on $k(x^2,x^3)=k(x)$
Such $F, G$ do exist, unless I missed a criterion.
One possibility is to take $F(T)$ a polynomial such that $F(h)$ is never zero, such as $T^3 + x^2$, and then $G(T) = x F(T)$. Then you can use the ...
- 148k
1
vote
Accepted
Using computer algebra to check if a family of algebras are pair-wise non-isomorphic
There's a practical computer-algebra way to determine whether
(a) they're all isomorphic up to finitely many exceptions, or
(b) there "(bounded finite)-to-one" non-isomorphic, i.e., there ...
- 63.9k
1
vote
Resultants for compactly represented product form polynomials?
Too long for a comment so I am posting this as an answer.
It would help if you said precisely what notion of resultant you are using and which variables are being eliminated. There are at least two, ...
1
vote
Efficient algorithm to compute resultants of sparse polynomials?
See the oevre of I. Emiris:
Emiris, Ioannis Z.; Pan, Victor Y., Improved algorithms for computing determinants and resultants, J. Complexity 21, No. 1, 43-71 (2005). ZBL1101.68981.
Jeronimo, ...
- 96.4k
Only top scored, non community-wiki answers of a minimum length are eligible