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Computing locus of points with positive dimensional fibers
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I became interested in the following problems by studying kinematic configuration spaces:
Let
$C=V(f_1,ldots,f_r)$ where $f_iin mathbbR[vecx,vecy]$,
$J=V(g_1,ldots,g_s)$ where $g_iin mathbbR[vecx]$,
$p:Cto J$ be a surjective open polynomial map.
Find $h_1,ldots, h_kin mathbbR[vec x]$ such that
$$
S:= vec xin J : dim p^-1(vec x) >0 = V(h_1,ldots, h_k)
$$
By the upper-semicontinuity of fiber dimension one can show that the set of points $vec c=(vec x,vec y)in C$ such that $dim p^-1(p(vec c)) = 0$ is Zariski open. Since $p$ is an open surjection it follows that the set of $vec xin J$ with finite fibers is Zariski open in $J$. Thus the locus of points in $J$ with positive dimensional fibers must be a subvariety of $J$. This justifies the existence of $h_1,...,h_k$.
Is there an algorithmic way to determine a generating set for $S$ that could be implemented in a computer algebra system?
I imagine this will involve some combination of things like Jacobians, rank, determinants and Gröbner basis. If it helps, in my case the map $p$ is the projection $p(vec x,vec y)=vec x$ restricted to $C$.
Similarly, consider
$$
S_d:= vec xin J : dim p^-1(vec x) geq d.
$$
Are there practical algorithmic methods to determine a generating set for $S_d$?
algebraic-geometry computational-geometry computational-algebra real-algebraic-geometry
asked 2 days ago
Christian Bueno
617314
add a comment |Â
up vote
1
down vote
favorite
I became interested in the following problems by studying kinematic configuration spaces:
Let
$C=V(f_1,ldots,f_r)$ where $f_iin mathbbR[vecx,vecy]$,
$J=V(g_1,ldots,g_s)$ where $g_iin mathbbR[vecx]$,
$p:Cto J$ be a surjective open polynomial map.
Find $h_1,ldots, h_kin mathbbR[vec x]$ such that
$$
S:= vec xin J : dim p^-1(vec x) >0 = V(h_1,ldots, h_k)
$$
By the upper-semicontinuity of fiber dimension one can show that the set of points $vec c=(vec x,vec y)in C$ such that $dim p^-1(p(vec c)) = 0$ is Zariski open. Since $p$ is an open surjection it follows that the set of $vec xin J$ with finite fibers is Zariski open in $J$. Thus the locus of points in $J$ with positive dimensional fibers must be a subvariety of $J$. This justifies the existence of $h_1,...,h_k$.
Is there an algorithmic way to determine a generating set for $S$ that could be implemented in a computer algebra system?
I imagine this will involve some combination of things like Jacobians, rank, determinants and Gröbner basis. If it helps, in my case the map $p$ is the projection $p(vec x,vec y)=vec x$ restricted to $C$.
Similarly, consider
$$
S_d:= vec xin J : dim p^-1(vec x) geq d.
$$
Are there practical algorithmic methods to determine a generating set for $S_d$?
algebraic-geometry computational-geometry computational-algebra real-algebraic-geometry
asked 2 days ago
Christian Bueno
617314
add a comment |Â
up vote
1
down vote
favorite
up vote
1
down vote
favorite
I became interested in the following problems by studying kinematic configuration spaces:
Let
$C=V(f_1,ldots,f_r)$ where $f_iin mathbbR[vecx,vecy]$,
$J=V(g_1,ldots,g_s)$ where $g_iin mathbbR[vecx]$,
$p:Cto J$ be a surjective open polynomial map.
Find $h_1,ldots, h_kin mathbbR[vec x]$ such that
$$
S:= vec xin J : dim p^-1(vec x) >0 = V(h_1,ldots, h_k)
$$
By the upper-semicontinuity of fiber dimension one can show that the set of points $vec c=(vec x,vec y)in C$ such that $dim p^-1(p(vec c)) = 0$ is Zariski open. Since $p$ is an open surjection it follows that the set of $vec xin J$ with finite fibers is Zariski open in $J$. Thus the locus of points in $J$ with positive dimensional fibers must be a subvariety of $J$. This justifies the existence of $h_1,...,h_k$.
Is there an algorithmic way to determine a generating set for $S$ that could be implemented in a computer algebra system?
I imagine this will involve some combination of things like Jacobians, rank, determinants and Gröbner basis. If it helps, in my case the map $p$ is the projection $p(vec x,vec y)=vec x$ restricted to $C$.
Similarly, consider
$$
S_d:= vec xin J : dim p^-1(vec x) geq d.
$$
Are there practical algorithmic methods to determine a generating set for $S_d$?
algebraic-geometry computational-geometry computational-algebra real-algebraic-geometry
asked 2 days ago
Christian Bueno
617314
I became interested in the following problems by studying kinematic configuration spaces:
Let
$C=V(f_1,ldots,f_r)$ where $f_iin mathbbR[vecx,vecy]$,
$J=V(g_1,ldots,g_s)$ where $g_iin mathbbR[vecx]$,
$p:Cto J$ be a surjective open polynomial map.
Find $h_1,ldots, h_kin mathbbR[vec x]$ such that
$$
S:= vec xin J : dim p^-1(vec x) >0 = V(h_1,ldots, h_k)
$$
By the upper-semicontinuity of fiber dimension one can show that the set of points $vec c=(vec x,vec y)in C$ such that $dim p^-1(p(vec c)) = 0$ is Zariski open. Since $p$ is an open surjection it follows that the set of $vec xin J$ with finite fibers is Zariski open in $J$. Thus the locus of points in $J$ with positive dimensional fibers must be a subvariety of $J$. This justifies the existence of $h_1,...,h_k$.
Is there an algorithmic way to determine a generating set for $S$ that could be implemented in a computer algebra system?
I imagine this will involve some combination of things like Jacobians, rank, determinants and Gröbner basis. If it helps, in my case the map $p$ is the projection $p(vec x,vec y)=vec x$ restricted to $C$.
Similarly, consider
$$
S_d:= vec xin J : dim p^-1(vec x) geq d.
$$
Are there practical algorithmic methods to determine a generating set for $S_d$?
algebraic-geometry computational-geometry computational-algebra real-algebraic-geometry
asked 2 days ago
Christian Bueno
617314
asked 2 days ago
Christian Bueno
617314
asked 2 days ago
Christian Bueno
617314
asked 2 days ago
Christian Bueno
617314
617314
add a comment |Â
add a comment |Â
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