Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: Under the assumption $P=\Sigma_2^p$,  prove a new variant of the Union Theorem of McCreight and Meyer for the class $\Sigma_2^p$. This yields a union function $F$ which is computable in time $F(n)^c$ for some constant $c$ and satisfies $P=DTIME(F)=\Sigma_2(F)=\Sigma_2^p$ with respect to a subfamily $(\tilde{S}_i)$ of $\Sigma_2$-machines.  show that this subfamily does not change the complexity classes $P$ and $\Sigma_2^p$. Moreover, a padding construction shows that this also implies $DTIME(F^c)=\Sigma_2(F^c)$. On the other hand,  prove a variant of Gupta's result who shod that $DTIME(t)\subsetneq\Sigma_2(t)$ for time-constructible functions $t(n)$. Our variant of this result holds with respect to the subfamily $(\tilde{S}_i)$ of $\Sigma_2$-machines.  show that these two results contradict each other. Hence the assumption $P=\Sigma_2^p$ cannot hold.
['Mathias Hauptmann']
I am writing a research paper on a idea On Alternation and the Union Theorem
[{"from": "human", "value": "I am writing a research paper on a idea On Alternation and the Union Theorem"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: Under the assumption $P=\\Sigma_2^p$,  prove a new variant of the Union Theorem of McCreight and Meyer for the class $\\Sigma_2^p$. This yields a union function $F$ which is computable in time $F(n)^c$ for some constant $c$ and satisfies $P=DTIME(F)=\\Sigma_2(F)=\\Sigma_2^p$ with respect to a subfamily $(\\tilde{S}_i)$ of $\\Sigma_2$-machines.  show that this subfamily does not change the complexity classes $P$ and $\\Sigma_2^p$. Moreover, a padding construction shows that this also implies $DTIME(F^c)=\\Sigma_2(F^c)$. On the other hand,  prove a variant of Gupta's result who shod that $DTIME(t)\\subsetneq\\Sigma_2(t)$ for time-constructible functions $t(n)$. Our variant of this result holds with respect to the subfamily $(\\tilde{S}_i)$ of $\\Sigma_2$-machines.  show that these two results contradict each other. Hence the assumption $P=\\Sigma_2^p$ cannot hold."}]