Non-existence of a generic solution to system of nonlinear equations
$begingroup$
I have the following system of nonlinear equations:
$f_1(x_1,..,x_m,y) =0$
$...$
$f_n(x_1,..,x_m,y) =0$
where $f_i(cdot)$ is a nonlinear, (infinitely) differentiable equation (but not polynomial), $x_1,..,x_m$ are the unknowns, $y$ is a vector of parameters, and $n>mgeq 4$, so the number of equations ($n$) is bigger than the number of unknowns ($m$).
I want to prove either that a solution to this system does not exist, or if a solution does exist, then it is unstable in the sense that if $y$ is perturbed, then a solution to this system ceases to exist.
I don't even know where to start looking for a theorem in that direction, so any help is greatly appreciated! Cheers!
algebraic-topology systems-of-equations nonlinear-system implicit-function-theorem
edited Dec 6 '18 at 2:54
KReiser
9,34721435
asked Dec 5 '18 at 22:32
George GeorgiadisGeorge Georgiadis
368
$endgroup$
add a comment |
$begingroup$
I have the following system of nonlinear equations:
$f_1(x_1,..,x_m,y) =0$
$...$
$f_n(x_1,..,x_m,y) =0$
where $f_i(cdot)$ is a nonlinear, (infinitely) differentiable equation (but not polynomial), $x_1,..,x_m$ are the unknowns, $y$ is a vector of parameters, and $n>mgeq 4$, so the number of equations ($n$) is bigger than the number of unknowns ($m$).
I want to prove either that a solution to this system does not exist, or if a solution does exist, then it is unstable in the sense that if $y$ is perturbed, then a solution to this system ceases to exist.
I don't even know where to start looking for a theorem in that direction, so any help is greatly appreciated! Cheers!
algebraic-topology systems-of-equations nonlinear-system implicit-function-theorem
edited Dec 6 '18 at 2:54
KReiser
9,34721435
asked Dec 5 '18 at 22:32
George GeorgiadisGeorge Georgiadis
368
$endgroup$
$begingroup$
Retagged with algebraic-topology instead of algebraic-geometry, as this is more appropriate for the problem. You may want to have a look at Sard's theorem and reinterpreting a solution of your system of equations as the intersection of level sets.
$endgroup$
– KReiser
Dec 6 '18 at 2:56
add a comment |
$begingroup$
I have the following system of nonlinear equations:
$f_1(x_1,..,x_m,y) =0$
$...$
$f_n(x_1,..,x_m,y) =0$
where $f_i(cdot)$ is a nonlinear, (infinitely) differentiable equation (but not polynomial), $x_1,..,x_m$ are the unknowns, $y$ is a vector of parameters, and $n>mgeq 4$, so the number of equations ($n$) is bigger than the number of unknowns ($m$).
I want to prove either that a solution to this system does not exist, or if a solution does exist, then it is unstable in the sense that if $y$ is perturbed, then a solution to this system ceases to exist.
I don't even know where to start looking for a theorem in that direction, so any help is greatly appreciated! Cheers!
algebraic-topology systems-of-equations nonlinear-system implicit-function-theorem
edited Dec 6 '18 at 2:54
KReiser
9,34721435
asked Dec 5 '18 at 22:32
George GeorgiadisGeorge Georgiadis
368
$endgroup$
I have the following system of nonlinear equations:
$f_1(x_1,..,x_m,y) =0$
$...$
$f_n(x_1,..,x_m,y) =0$
where $f_i(cdot)$ is a nonlinear, (infinitely) differentiable equation (but not polynomial), $x_1,..,x_m$ are the unknowns, $y$ is a vector of parameters, and $n>mgeq 4$, so the number of equations ($n$) is bigger than the number of unknowns ($m$).
I want to prove either that a solution to this system does not exist, or if a solution does exist, then it is unstable in the sense that if $y$ is perturbed, then a solution to this system ceases to exist.
I don't even know where to start looking for a theorem in that direction, so any help is greatly appreciated! Cheers!
algebraic-topology systems-of-equations nonlinear-system implicit-function-theorem
algebraic-topology systems-of-equations nonlinear-system implicit-function-theorem
edited Dec 6 '18 at 2:54
KReiser
9,34721435
asked Dec 5 '18 at 22:32
George GeorgiadisGeorge Georgiadis
368
edited Dec 6 '18 at 2:54
KReiser
9,34721435
asked Dec 5 '18 at 22:32
George GeorgiadisGeorge Georgiadis
368
edited Dec 6 '18 at 2:54
KReiser
9,34721435
edited Dec 6 '18 at 2:54
KReiser
9,34721435
edited Dec 6 '18 at 2:54
KReiser
9,34721435
9,34721435
asked Dec 5 '18 at 22:32
George GeorgiadisGeorge Georgiadis
368
asked Dec 5 '18 at 22:32
George GeorgiadisGeorge Georgiadis
368
asked Dec 5 '18 at 22:32
George GeorgiadisGeorge Georgiadis
368
368
$begingroup$
Retagged with algebraic-topology instead of algebraic-geometry, as this is more appropriate for the problem. You may want to have a look at Sard's theorem and reinterpreting a solution of your system of equations as the intersection of level sets.
$endgroup$
– KReiser
Dec 6 '18 at 2:56
add a comment |
$begingroup$
Retagged with algebraic-topology instead of algebraic-geometry, as this is more appropriate for the problem. You may want to have a look at Sard's theorem and reinterpreting a solution of your system of equations as the intersection of level sets.
$endgroup$
– KReiser
Dec 6 '18 at 2:56
$begingroup$
Retagged with algebraic-topology instead of algebraic-geometry, as this is more appropriate for the problem. You may want to have a look at Sard's theorem and reinterpreting a solution of your system of equations as the intersection of level sets.
$endgroup$
– KReiser
Dec 6 '18 at 2:56
$begingroup$
Retagged with algebraic-topology instead of algebraic-geometry, as this is more appropriate for the problem. You may want to have a look at Sard's theorem and reinterpreting a solution of your system of equations as the intersection of level sets.
$endgroup$
– KReiser
Dec 6 '18 at 2:56
add a comment |
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$begingroup$
Retagged with algebraic-topology instead of algebraic-geometry, as this is more appropriate for the problem. You may want to have a look at Sard's theorem and reinterpreting a solution of your system of equations as the intersection of level sets.
$endgroup$
– KReiser
Dec 6 '18 at 2:56