We call any consistent and sufficiently powerful formal theory that enables to algorithmically verify whether a text is a proof \textbf{algorithmically verifiable mathematics} (av-mathematics). We study the question whether nondeterminism is more powerful than determinism for polynomial time computations in the framework of av-mathematics. Our main results are as follows. \\
"\P \subsetneq \NP or for any deterministic, polynomial time compression algorithm A there exists a nondeterministic, polynomial time compression machine M that reduces infinitely many binary strings logarithmically stronger than A." \\
"\P \subsetneq \NP or f-time resource bounded Kolmogorov complexity of any binary string x can be computed in deterministic polynomial time for each polynomial, time constructible function f."\\
For computing models with "efficient" interpreters we prove the following theorem:\\
"For each polynomial, time constructible function f, \TIMEf \subsetneq \NTIMEf or one can essentially stronger compress words nondeterministically in time \Oh{f(n)} than deterministically in time f(n)."

strengthening some of the results

We call any consistent and sufficiently powerful formal theory that enables to algorithmically in polynomial time verify whether a text is a proof \textbf{efficiently verifiable mathematics} (ev-mathematics). We study the question whether nondeterminism is more powerful than determinism for polynomial time computations in the framework of ev-mathematics. Our main results are as follows. \\
"\P \subsetneq \NP or for any deterministic, polynomial time compression algorithm A there exists a nondeterministic, polynomial time compression machine M that reduces infinitely many binary strings logarithmically stronger than A." \\
"\P \subsetneq \NP or f-time resource bounded Kolmogorov complexity of any binary string x can be computed in deterministic polynomial time for each polynomial time constructible function f."