Does there exist $f$ continuous map from $\mathbb C^4$ to $\mathbb R$ such that  :

i)there exists four distinct complex numbers $a,b,c,d$, s.t.$f(a,b,c,d).f(b,c,d,a)<0$

ii)for any $(x,y,z,t)\in \mathbb C^4$, $f(x,y,z,t)=0$ implies $x,y,z,t$ are the corners of a square in the complex plan?

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this is related answer to https://en.m.wikipedia.org/wiki/Inscribed_square_problem

Indeed, let's say that a polygone defined by $(a,b,c,d)$ is $f$-good if it satisfies the condition i), then an approche to solve the inscribed square problem can be to prove that there exists $f$ such that any Jordan curve contain a $f$-good polygone.

Does there exist a continuous map $f$ from $\mathbb C^4$ to $\mathbb R$ such that:

i) there exists four distinct complex numbers $a$, $b$, $c$, $d$, s.t. $f(a,b,c,d)f(b,c,d,a)<0$

ii) for every $(x,y,z,t)\in \mathbb C^4$, $f(x,y,z,t)=0$ implies that $x$, $y$, $z$, $t$ are the corners of a square in the complex plane?

This is related to the inscribed square problem.

Indeed, let's say that a polygon defined by $(a,b,c,d)$ is $f$-good if it satisfies the condition i), then an approach to solve the inscribed square problem can be to prove that there exists $f$ such that any Jordan curve contain a $f$-good polygon.

Does there exist $f$ continuous map from $\mathbb C^4$ to $\mathbb R$ such that :

i)there exists four distinct complex numbers $a,b,c,d$, s.t.$f(a,b,c,d).f(b,c,d,a)<0$

ii)for any $(x,y,z,t)\in \mathbb C^4$, $f(x,y,z,t)=0$ implies $x,y,z$ and $t$ are the vertices of square in the complex plan?

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Same question with $f$ as a polynome*, or seen as a polynome with 8 real entries.

*(that justifies the tag "model theory")

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this is related answer to https://en.m.wikipedia.org/wiki/Inscribed_square_problem

Does there exist $f$ continuous map from $\mathbb C^4$ to $\mathbb R$ such that :

i)there exists four distinct complex numbers $a,b,c,d$, s.t.$f(a,b,c,d).f(b,c,d,a)<0$

ii)for any $(x,y,z,t)\in \mathbb C^4$, $f(x,y,z,t)=0$ implies $x,y,z,t$ are the corners of a square in the complex plan?

-‐--------------------

this is related answer to https://en.m.wikipedia.org/wiki/Inscribed_square_problem

Indeed, let's say that a polygone defined by $(a,b,c,d)$ is $f$-good if it satisfies the condition i), then an approche to solve the inscribed square problem can be to prove that there exists $f$ such that any Jordan curve contain a $f$-good polygone.

Does there exist $f$ continuous map from $\mathbb C^4$ to $\mathbb R$ such that :

i)there exists four distinct complex numbers $a,b,c,d$, s.t.$f(a,b,c,d).f(b,c,d,a)<0$

ii)for any $(x,y,z,t)\in \mathbb C^4$, $f(x,y,z,t)=0$ implies $(x,y,z,t)$ is a square?

‐-‐--------------

Same question with $f$ as a polynome*, or seen as a polynome with 8 real entries.

*(that justifies the tag "model theory")

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A positive answer to this question would give a positive answer to : is there in the image of any continuous injection from the circle in the plan, four points that form a square ? (i don't know if this problem has been solved)

Does there exist $f$ continuous map from $\mathbb C^4$ to $\mathbb R$ such that :

i)there exists four distinct complex numbers $a,b,c,d$, s.t.$f(a,b,c,d).f(b,c,d,a)<0$

ii)for any $(x,y,z,t)\in \mathbb C^4$, $f(x,y,z,t)=0$ implies $x,y,z$ and $t$ are the vertices of square in the complex plan?

‐-‐--------------

Same question with $f$ as a polynome*, or seen as a polynome with 8 real entries.

*(that justifies the tag "model theory")

-‐--------------------

this is related answer to https://en.m.wikipedia.org/wiki/Inscribed_square_problem