Return to Answer

There are other strange assertions in the paper. For instance p. 131. The author wants to prove (Theorem 7.3) that that a certain meromorphic function $P/f$ has no pole, where the holomorphic function $f$ has (at least) a simple zero at $\omega\in \mathbb C$, and $P$ is a polynomial. The proof is very odd:First of all, it is never proved that $U=P/f$ has no pole. Secondly, $U$ is treated... like it was already proved that it has... no pole. More precisely, the author considers $U$ as a formal series in $X$ (like $1/X= 1/z$ is a Laurent series). Then he derives $P=Uf$ and gets $P'=U'f +Uf'$. But now he writes "specializing the formal variable $X$ to the complex variable $z$" and gets $$P'(z) = U'(z)f(z) +U(z)f'(z),$$ forgetting that this formal hocus-pocus does not make $U(z)$ (and $U'(z)$) a defined complex number! He then makes $z=\omega$ and obtains $P'(\omega) = U(\omega) f'(\omega)$, forgetting that $U$ is still a formal series. In fact this argument holds for $P=1$ and $f=z$. It would then be proved that $U(z) =1/z$... is holomorphic at $0$!

Added: there are (at least) two other places where the same argument is used : p110 and p125. The 'fracturability' defined p. 4 is this kind of strange factorization of polynomials, like $1$ is factorized as $1=(1/z)z$...

There are other strange assertions in the paper. For instance p. 131. The author wants to prove (Theorem 7.3) that that a certain meromorphic function $P/f$ has no pole, where the holomorphic function $f$ has (at least) a simple zero at $\omega\in \mathbb C$, and $P$ is a polynomial. The proof is very odd:First of all, it is never proved that $U=P/f$ has no pole. Secondly, $U$ is treated... like it was already proved that it has... no pole. More precisely, the author considers $U$ as a formal series in $X$ (like $1/X= 1/z$ is a Laurent series). Then he derives $P=Uf$ and gets $P'=U'f +Uf'$. But now he writes "specializing the formal variable $X$ to the complex variable $z$" and gets $$P'(z) = U'(z)f(z) +U(z)f'(z),$$ forgetting that this formal hocus-pocus does not make $U(z)$ (and $U'(z)$) a defined complex number! He then makes $z=\omega$ and obtains $P'(\omega) = U(\omega) f'(\omega)$, forgetting that $U$ is still a formal series. In fact this argument holds for $P=1$ and $f=z$. It would then be proved that $U(z) =1/z$... is holomorphic at $0$!

Added: there are (at least) two other places where the same argument is used : p110 and p125. The 'fracturability' defined p. 4 is this kind of strange factorization of polynomials, like $1$ is factorized as $1=(1/z)z$, and $1/z$ becomes a holomorphic function...

There are other strange assertions in the paper. For instance p. 131. The author wants to prove (Theorem 7.3) that that a certain meromorphic function $P/f$ has no pole, where the holomorphic function $f$ has (at least) a simple zero at $\omega\in \mathbb C$, and $P$ is a polynomial. The proof is very odd:First of all, it is never proved that $U=P/f$ has no pole. Secondly, $U$ is treated... like it was already proved that it has... no pole. More precisely, the author considers $U$ as a formal series in $X$ (like $1/X= 1/z$ is a Laurent series). Then he derives $P=Uf$ and gets $P'=U'f +Uf'$. But now he writes "specializing the formal variable $X$ to the complex variable $z$" and gets $$P'(z) = U'(z)f(z) +U(z)f'(z),$$ forgetting that this formal hocus-pocus does not make $U(z)$ (and $U'(z)$) a defined complex number! He then makes $z=\omega$ and obtains $P'(\omega) = U(\omega) f'(\omega)$, forgetting that $U$ is still a formal series. In fact this argument holds for $P=1$ and $f=z$. It would then be proved that $U(z) =1/z$... is holomorphic at $0$!

There are other strange assertions in the paper. For instance p. 131. The author wants to prove (Theorem 7.3) that that a certain meromorphic function $P/f$ has no pole, where the holomorphic function $f$ has (at least) a simple zero at $\omega\in \mathbb C$, and $P$ is a polynomial. The proof is very odd:First of all, it is never proved that $U=P/f$ has no pole. Secondly, $U$ is treated... like it was already proved that it has... no pole. More precisely, the author considers $U$ as a formal series in $X$ (like $1/X= 1/z$ is a Laurent series). Then he derives $P=Uf$ and gets $P'=U'f +Uf'$. But now he writes "specializing the formal variable $X$ to the complex variable $z$" and gets $$P'(z) = U'(z)f(z) +U(z)f'(z),$$ forgetting that this formal hocus-pocus does not make $U(z)$ (and $U'(z)$) a defined complex number! He then makes $z=\omega$ and obtains $P'(\omega) = U(\omega) f'(\omega)$, forgetting that $U$ is still a formal series. In fact this argument holds for $P=1$ and $f=z$. It would then be proved that $U(z) =1/z$... is holomorphic at $0$!

Added: there are (at least) two other places where the same argument is used : p110 and p125. The 'fracturability' defined p. 4 is this kind of strange factorization of polynomials, like $1$ is factorized as $1=(1/z)z$...

There are other strange assertions in the paper. For instance p. 131. The author wants to prove (Theorem 7.3) that that a certain meromorphic function $P/f$ has no pole, where the holomorphic function $f$ has (at least) a simple zero at $\omega\in \mathbb C$, and $P$ is a polynomial. The proof is very odd:First of all, it is never proved that $U=P/f$ is has no pole. Secondly, $U$ is treated... like it was already proved that it has... no pole. More precisely, the author considers $U$ as a formal series in $X$ (like $1/X= 1/z$ is a Laurent series). Then he derives $P=Uf$ and gets $P'=U'f +Uf'$. But now he writes "specializing the formal variable $X$ to the complex variable $z$" and gets $$P'(z) = U'(z)f(z) +U(z)f'(z),$$ forgetting that this formal hocus-pocus does not make $U(z)$ (and $U'(z)$) a defined complex number! He then makes $z=\omega$ and obtains $P'(\omega) = U(\omega) f'(\omega)$, forgetting that $U$ is still a formal series. In fact this argument holds for $P=1$ and $f=z$. It would then be proved that $U(z) =1/z$... is holomorphic at $0$!

There are other strange assertions in the paper. For instance p. 131. The author wants to prove (Theorem 7.3) that that a certain meromorphic function $P/f$ has no pole, where the holomorphic function $f$ has (at least) a simple zero at $\omega\in \mathbb C$, and $P$ is a polynomial. The proof is very odd:First of all, it is never proved that $U=P/f$ has no pole. Secondly, $U$ is treated... like it was already proved that it has... no pole. More precisely, the author considers $U$ as a formal series in $X$ (like $1/X= 1/z$ is a Laurent series). Then he derives $P=Uf$ and gets $P'=U'f +Uf'$. But now he writes "specializing the formal variable $X$ to the complex variable $z$" and gets $$P'(z) = U'(z)f(z) +U(z)f'(z),$$ forgetting that this formal hocus-pocus does not make $U(z)$ (and $U'(z)$) a defined complex number! He then makes $z=\omega$ and obtains $P'(\omega) = U(\omega) f'(\omega)$, forgetting that $U$ is still a formal series. In fact this argument holds for $P=1$ and $f=z$. It would then be proved that $U(z) =1/z$... is holomorphic at $0$!