Applications of polynomial inverse root construction
If $p(x) = sum_{j=0}^n a_j x^j$ is a polynomial of degree $n$, and $r_j$ are its roots, there are usually not very explicit relationships between transformations of the coefficients $a_j$ and the roots $r_j$. One very cute exception is the map
$$
sum_{j=0}^n a_j x^j
mapsto
sum_{j=0}^n a_{n-j} x^j
$$
which transforms the roots as $r_j mapsto 1/r_j$ when $r_j not= 0$, and converts a root $0$ of multiplicity $k$ to a root $0$ of multiplicity $n-k$.
The proof of this is to move to a field where $p$ splits into linear factors, notice that the transformation is also given by the formal multiplication $p(x) mapsto x^n p(1/x)$, and inspect what that does to the factorization of $p$ into linear factors.
This is certainly very classical, but does anyone know a reference for this fact or, more interestingly, some applications of it?
polynomials reference-request
asked Nov 29 at 6:28
Gunnar Þór Magnússon
3,9631326
add a comment |
If $p(x) = sum_{j=0}^n a_j x^j$ is a polynomial of degree $n$, and $r_j$ are its roots, there are usually not very explicit relationships between transformations of the coefficients $a_j$ and the roots $r_j$. One very cute exception is the map
$$
sum_{j=0}^n a_j x^j
mapsto
sum_{j=0}^n a_{n-j} x^j
$$
which transforms the roots as $r_j mapsto 1/r_j$ when $r_j not= 0$, and converts a root $0$ of multiplicity $k$ to a root $0$ of multiplicity $n-k$.
The proof of this is to move to a field where $p$ splits into linear factors, notice that the transformation is also given by the formal multiplication $p(x) mapsto x^n p(1/x)$, and inspect what that does to the factorization of $p$ into linear factors.
This is certainly very classical, but does anyone know a reference for this fact or, more interestingly, some applications of it?
polynomials reference-request
asked Nov 29 at 6:28
Gunnar Þór Magnússon
3,9631326
add a comment |
If $p(x) = sum_{j=0}^n a_j x^j$ is a polynomial of degree $n$, and $r_j$ are its roots, there are usually not very explicit relationships between transformations of the coefficients $a_j$ and the roots $r_j$. One very cute exception is the map
$$
sum_{j=0}^n a_j x^j
mapsto
sum_{j=0}^n a_{n-j} x^j
$$
which transforms the roots as $r_j mapsto 1/r_j$ when $r_j not= 0$, and converts a root $0$ of multiplicity $k$ to a root $0$ of multiplicity $n-k$.
The proof of this is to move to a field where $p$ splits into linear factors, notice that the transformation is also given by the formal multiplication $p(x) mapsto x^n p(1/x)$, and inspect what that does to the factorization of $p$ into linear factors.
This is certainly very classical, but does anyone know a reference for this fact or, more interestingly, some applications of it?
polynomials reference-request
asked Nov 29 at 6:28
Gunnar Þór Magnússon
3,9631326
If $p(x) = sum_{j=0}^n a_j x^j$ is a polynomial of degree $n$, and $r_j$ are its roots, there are usually not very explicit relationships between transformations of the coefficients $a_j$ and the roots $r_j$. One very cute exception is the map
$$
sum_{j=0}^n a_j x^j
mapsto
sum_{j=0}^n a_{n-j} x^j
$$
which transforms the roots as $r_j mapsto 1/r_j$ when $r_j not= 0$, and converts a root $0$ of multiplicity $k$ to a root $0$ of multiplicity $n-k$.
The proof of this is to move to a field where $p$ splits into linear factors, notice that the transformation is also given by the formal multiplication $p(x) mapsto x^n p(1/x)$, and inspect what that does to the factorization of $p$ into linear factors.
This is certainly very classical, but does anyone know a reference for this fact or, more interestingly, some applications of it?
polynomials reference-request
polynomials reference-request
asked Nov 29 at 6:28
Gunnar Þór Magnússon
3,9631326
asked Nov 29 at 6:28
Gunnar Þór Magnússon
3,9631326
asked Nov 29 at 6:28
Gunnar Þór Magnússon
3,9631326
asked Nov 29 at 6:28
Gunnar Þór Magnússon
3,9631326
asked Nov 29 at 6:28
Gunnar Þór Magnússon
3,9631326
3,9631326
add a comment |
add a comment |
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