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Let ${\mathbf P}$ be the class of polynomial-time decision problems and$\mathbf{NP}$ be the class of nondeterministic polynomial time decisionproblems. We prove the following:  Theorem 3. The classes ${\mathbf P}$ and $\mathbf{NP}$ are equivalent. Thatis, ${\mathbf P}=\mathbf{NP}$. Theorem 3 gives a positive answer to thequestion $$\hbox{Does }{\mathbf P}=\mathbf{NP}?,$$ 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, fromwhich it follows that the $\mathbf{NP}$-complete problem of deciding theHilbert Nullstellensatz over $\mathbb{Z}_2$ belongs to the class ${\mathbf P}$.  Theorem 2. There is a constructive algorithm for deciding the HilbertNullstellensatz over $\mathbb{Z}_2$, where $\mathbb{Z}_2$ is the space of allcomplex numbers with integer real and imaginary parts. The number$s(n,m_{\sigma})$ of basic steps of the algorithm, where $n$ is the number ofvariables and $m_{\sigma}$ is the total length of input polynomials, satisfiesthe inequality \begin{eqnarray*} & & s(n,m_{\sigma}) \\ & \le &c_2\{m_{\sigma}^2\logm_{\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 absoluteconstant, $\{d_{\ell}\}_{\ell=1}^n$ 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, concerningpartial lengths and numbers of major sub-monomials it the natural order ofmonomials, defined in the body of the paper.

A polynomial-time algorithm for deciding the Hilbert Nullstellensatz over $\mathbb{Z}_2$. A proof of $\mathbf{P}=\mathbf{NP}$ hypothesis