Mixing
When do polynomial equations come from complexification?
$begingroup$
If $f(z) in mathbb{C}[z]$ is a polynomial of degree $d$, then it has $d$ complex zeros. Writing the complexification $$f(x+iy)=u(x,y)+iv(x,y)$$ we observe that the real polynomial system $u(x,y)=v(x,y)=0$ has $d$ real solutions (corresponding to the $d$ complex zeros of $f$).
Motivating question: if you are given $u(x,y),v(x,y)$ how could you recognize whether they came from a complexification $f(x+iy)$?
The answer is to check the Cauchy Riemann equations.
Actual (and more difficult) question: Suppose you are given new generators $g(x,y),h(x,y)$ for the ideal $langle u(x,y),v(x,y) rangle$ (where $u$ and $v$ are the real and imaginary parts of the complexification of a univariate polynomial).
Although the solutions $g=h=0$ are the same as $u=v=0$, you can check that $g(x,y),h(x,y)$ (generically) do not satisfy the Cauchy-Riemann equations.
Now how can one tell when $g(x,y),h(x,y)$ came about in this fashion? and thus be able to tell immediately that they have real common solutions?
complex-analysis polynomials real-algebraic-geometry
asked Jan 17 at 22:12
TaylorTaylor
412
$endgroup$
add a comment |
$begingroup$
If $f(z) in mathbb{C}[z]$ is a polynomial of degree $d$, then it has $d$ complex zeros. Writing the complexification $$f(x+iy)=u(x,y)+iv(x,y)$$ we observe that the real polynomial system $u(x,y)=v(x,y)=0$ has $d$ real solutions (corresponding to the $d$ complex zeros of $f$).
Motivating question: if you are given $u(x,y),v(x,y)$ how could you recognize whether they came from a complexification $f(x+iy)$?
The answer is to check the Cauchy Riemann equations.
Actual (and more difficult) question: Suppose you are given new generators $g(x,y),h(x,y)$ for the ideal $langle u(x,y),v(x,y) rangle$ (where $u$ and $v$ are the real and imaginary parts of the complexification of a univariate polynomial).
Although the solutions $g=h=0$ are the same as $u=v=0$, you can check that $g(x,y),h(x,y)$ (generically) do not satisfy the Cauchy-Riemann equations.
Now how can one tell when $g(x,y),h(x,y)$ came about in this fashion? and thus be able to tell immediately that they have real common solutions?
complex-analysis polynomials real-algebraic-geometry
asked Jan 17 at 22:12
TaylorTaylor
412
$endgroup$
add a comment |
$begingroup$
If $f(z) in mathbb{C}[z]$ is a polynomial of degree $d$, then it has $d$ complex zeros. Writing the complexification $$f(x+iy)=u(x,y)+iv(x,y)$$ we observe that the real polynomial system $u(x,y)=v(x,y)=0$ has $d$ real solutions (corresponding to the $d$ complex zeros of $f$).
Motivating question: if you are given $u(x,y),v(x,y)$ how could you recognize whether they came from a complexification $f(x+iy)$?
The answer is to check the Cauchy Riemann equations.
Actual (and more difficult) question: Suppose you are given new generators $g(x,y),h(x,y)$ for the ideal $langle u(x,y),v(x,y) rangle$ (where $u$ and $v$ are the real and imaginary parts of the complexification of a univariate polynomial).
Although the solutions $g=h=0$ are the same as $u=v=0$, you can check that $g(x,y),h(x,y)$ (generically) do not satisfy the Cauchy-Riemann equations.
Now how can one tell when $g(x,y),h(x,y)$ came about in this fashion? and thus be able to tell immediately that they have real common solutions?
complex-analysis polynomials real-algebraic-geometry
asked Jan 17 at 22:12
TaylorTaylor
412
$endgroup$
If $f(z) in mathbb{C}[z]$ is a polynomial of degree $d$, then it has $d$ complex zeros. Writing the complexification $$f(x+iy)=u(x,y)+iv(x,y)$$ we observe that the real polynomial system $u(x,y)=v(x,y)=0$ has $d$ real solutions (corresponding to the $d$ complex zeros of $f$).
Motivating question: if you are given $u(x,y),v(x,y)$ how could you recognize whether they came from a complexification $f(x+iy)$?
The answer is to check the Cauchy Riemann equations.
Actual (and more difficult) question: Suppose you are given new generators $g(x,y),h(x,y)$ for the ideal $langle u(x,y),v(x,y) rangle$ (where $u$ and $v$ are the real and imaginary parts of the complexification of a univariate polynomial).
Although the solutions $g=h=0$ are the same as $u=v=0$, you can check that $g(x,y),h(x,y)$ (generically) do not satisfy the Cauchy-Riemann equations.
Now how can one tell when $g(x,y),h(x,y)$ came about in this fashion? and thus be able to tell immediately that they have real common solutions?
complex-analysis polynomials real-algebraic-geometry
complex-analysis polynomials real-algebraic-geometry
asked Jan 17 at 22:12
TaylorTaylor
412
asked Jan 17 at 22:12
TaylorTaylor
412
asked Jan 17 at 22:12
TaylorTaylor
412
asked Jan 17 at 22:12
TaylorTaylor
412
asked Jan 17 at 22:12
TaylorTaylor
412
412
add a comment |
add a comment |
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