Algorithm to find relations between polynomials
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Let $p_1,ldots,p_kinmathbbC[x_1,ldots,x_n]$ be polynomials. Is there an algorithm to compute the ideal of relations between them?
More precisely, to find a set of generators for the kernel of the ring homomorphism
$$mathbbC[y_1,ldots,y_k]to mathbbC[x_1,ldots,x_n],quad y_imapsto p_i(x_1,ldots,x_n).$$
(This enables us to compute $mathbbC[p_1,ldots,p_k]$).
Example: Consider
$$x_1x_2,x_3x_4,x_1x_3,x_2x_4inmathbbC[x_1,x_2,x_3,x_4].$$
One obvious relation between them is
$$(x_1x_2)(x_3x_4)=(x_1x_3)(x_2x_4).$$
Thus, if
$$varphi:mathbbC[y_1,y_2,y_3,y_4]tomathbbC[x_1,x_2,x_3,x_4]$$
is the homomorphism defined by
$$y_1mapsto x_1x_2,quad y_2mapsto x_3x_4,quad y_3mapsto x_1x_3,quad y_4mapsto x_2x_4,$$
then $y_1y_2-y_3y_4inker varphi$. It is easy to see that $kervarphi$ is in fact generated by $y_1y_2-y_3y_4$, so
$$mathbbC[x_1x_2,x_3x_4,x_1x_3,x_2x_4]congmathbbC[y_1,y_2,y_3,y_3]/(y_1y_2-y_3y_4).$$
(I posted a similar question in mathematica.stackexchange.)
abstract-algebra polynomials ring-theory algorithms polynomial-rings
asked Sep 2 at 11:12
Simon Parker
1,4002917
add a comment |Â
up vote
3
down vote
favorite
Let $p_1,ldots,p_kinmathbbC[x_1,ldots,x_n]$ be polynomials. Is there an algorithm to compute the ideal of relations between them?
More precisely, to find a set of generators for the kernel of the ring homomorphism
$$mathbbC[y_1,ldots,y_k]to mathbbC[x_1,ldots,x_n],quad y_imapsto p_i(x_1,ldots,x_n).$$
(This enables us to compute $mathbbC[p_1,ldots,p_k]$).
Example: Consider
$$x_1x_2,x_3x_4,x_1x_3,x_2x_4inmathbbC[x_1,x_2,x_3,x_4].$$
One obvious relation between them is
$$(x_1x_2)(x_3x_4)=(x_1x_3)(x_2x_4).$$
Thus, if
$$varphi:mathbbC[y_1,y_2,y_3,y_4]tomathbbC[x_1,x_2,x_3,x_4]$$
is the homomorphism defined by
$$y_1mapsto x_1x_2,quad y_2mapsto x_3x_4,quad y_3mapsto x_1x_3,quad y_4mapsto x_2x_4,$$
then $y_1y_2-y_3y_4inker varphi$. It is easy to see that $kervarphi$ is in fact generated by $y_1y_2-y_3y_4$, so
$$mathbbC[x_1x_2,x_3x_4,x_1x_3,x_2x_4]congmathbbC[y_1,y_2,y_3,y_3]/(y_1y_2-y_3y_4).$$
(I posted a similar question in mathematica.stackexchange.)
abstract-algebra polynomials ring-theory algorithms polynomial-rings
asked Sep 2 at 11:12
Simon Parker
1,4002917
add a comment |Â
up vote
3
down vote
favorite
up vote
3
down vote
favorite
Let $p_1,ldots,p_kinmathbbC[x_1,ldots,x_n]$ be polynomials. Is there an algorithm to compute the ideal of relations between them?
More precisely, to find a set of generators for the kernel of the ring homomorphism
$$mathbbC[y_1,ldots,y_k]to mathbbC[x_1,ldots,x_n],quad y_imapsto p_i(x_1,ldots,x_n).$$
(This enables us to compute $mathbbC[p_1,ldots,p_k]$).
Example: Consider
$$x_1x_2,x_3x_4,x_1x_3,x_2x_4inmathbbC[x_1,x_2,x_3,x_4].$$
One obvious relation between them is
$$(x_1x_2)(x_3x_4)=(x_1x_3)(x_2x_4).$$
Thus, if
$$varphi:mathbbC[y_1,y_2,y_3,y_4]tomathbbC[x_1,x_2,x_3,x_4]$$
is the homomorphism defined by
$$y_1mapsto x_1x_2,quad y_2mapsto x_3x_4,quad y_3mapsto x_1x_3,quad y_4mapsto x_2x_4,$$
then $y_1y_2-y_3y_4inker varphi$. It is easy to see that $kervarphi$ is in fact generated by $y_1y_2-y_3y_4$, so
$$mathbbC[x_1x_2,x_3x_4,x_1x_3,x_2x_4]congmathbbC[y_1,y_2,y_3,y_3]/(y_1y_2-y_3y_4).$$
(I posted a similar question in mathematica.stackexchange.)
abstract-algebra polynomials ring-theory algorithms polynomial-rings
asked Sep 2 at 11:12
Simon Parker
1,4002917
Let $p_1,ldots,p_kinmathbbC[x_1,ldots,x_n]$ be polynomials. Is there an algorithm to compute the ideal of relations between them?
More precisely, to find a set of generators for the kernel of the ring homomorphism
$$mathbbC[y_1,ldots,y_k]to mathbbC[x_1,ldots,x_n],quad y_imapsto p_i(x_1,ldots,x_n).$$
(This enables us to compute $mathbbC[p_1,ldots,p_k]$).
Example: Consider
$$x_1x_2,x_3x_4,x_1x_3,x_2x_4inmathbbC[x_1,x_2,x_3,x_4].$$
One obvious relation between them is
$$(x_1x_2)(x_3x_4)=(x_1x_3)(x_2x_4).$$
Thus, if
$$varphi:mathbbC[y_1,y_2,y_3,y_4]tomathbbC[x_1,x_2,x_3,x_4]$$
is the homomorphism defined by
$$y_1mapsto x_1x_2,quad y_2mapsto x_3x_4,quad y_3mapsto x_1x_3,quad y_4mapsto x_2x_4,$$
then $y_1y_2-y_3y_4inker varphi$. It is easy to see that $kervarphi$ is in fact generated by $y_1y_2-y_3y_4$, so
$$mathbbC[x_1x_2,x_3x_4,x_1x_3,x_2x_4]congmathbbC[y_1,y_2,y_3,y_3]/(y_1y_2-y_3y_4).$$
(I posted a similar question in mathematica.stackexchange.)
abstract-algebra polynomials ring-theory algorithms polynomial-rings
abstract-algebra polynomials ring-theory algorithms polynomial-rings
asked Sep 2 at 11:12
Simon Parker
1,4002917
asked Sep 2 at 11:12
Simon Parker
1,4002917
asked Sep 2 at 11:12
Simon Parker
1,4002917
asked Sep 2 at 11:12
Simon Parker
1,4002917
asked Sep 2 at 11:12
Simon Parker
1,4002917
1,4002917
add a comment |Â
add a comment |Â
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