  Let ${\mathbf P}$ be the class of polynomial-time decision problems and
$\mathbf{NP}$ be the class of nondeterministic polynomial time decision
problems. We prove the following:
  Theorem 3. The classes ${\mathbf P}$ and $\mathbf{NP}$ are equivalent. That
is, ${\mathbf P}=\mathbf{NP}$. Theorem 3 gives a positive answer to the
question $$\hbox{Does }{\mathbf P}=\mathbf{NP}?,$$ see S. Cook, The
$\mathbf{P}$ versus $\mathbf{NP}$ problem, Official problem description,
www.claymath.org/millennium-problems. Crucial for its proof is Theorem 2, from
which it follows that the $\mathbf{NP}$-complete problem of deciding the
Hilbert Nullstellensatz over $\mathbb{Z}_2$ belongs to the class ${\mathbf P}$.
  Theorem 2. There is a constructive algorithm for deciding the Hilbert
Nullstellensatz over $\mathbb{Z}_2$, where $\mathbb{Z}_2$ is the space of all
complex numbers with integer real and imaginary parts. The number
$s(n,m_{\sigma})$ of basic steps of the algorithm, where $n$ is the number of
variables and $m_{\sigma}$ is the total length of input polynomials, satisfies
the inequality \begin{eqnarray*} & & s(n,m_{\sigma}) \\ & \le &
c_2\{m_{\sigma}^2\log
m_{\sigma}+\min\{[m_{\sigma}^{(1)}]^3,(d_1)^3\}+\sum_{\ell
=1}^{n-2}N^{(l)}\min\{[m_{\sigma}^{(\ell +1)}]^2,(d_{\ell +1})^2)\}\\ &&
+N^{(n-1)}\min \{m_{\sigma},d_n\} \} \end{eqnarray*} where $c_2$ is an absolute
constant, $\{d_{\ell}\}_{\ell=1}^n$ are the maximal partial degrees in
$\{z_{\ell}\}_{\ell=1}^n$, respectively, and the numbers $m_{\sigma}^{(\ell)}$
and $N^{(\ell)}$ are characteristics of the input polynomials, concerning
partial lengths and numbers of major sub-monomials it the natural order of
monomials, defined in the body of the paper.
