@ARTICLE{POLYNOM9.MIZ,
  AUTHOR = {P\k{a}k, Karol},
  TITLE = {Prime Representing Polynomial with 10 Unknowns},
  SECTION1 = {Preliminaries},
  SECTION2 = {More on Bags},
  SECTION3 = {Power of Multivariate Polynomial},
  SECTION4 = {Substitution in Multivariate Polynomials},
  SECTION5 = {Set of Variables Used in Multivariate Polynomial},
  SECTION6 = {Polynomial Without the Last Variable},
  SECTION7 = {Jpolynom},
  SECTION8 = {Prime Representing Polynomial Construction},
  DAY = {27},
  MONTH = {December},
  YEAR = {2022},
  ADDRESS1 = {Institute of Computer Science\\University of Bia{\l}ystok\\ 
        Konstantego Cio{\l}kowskiego 1M, 15-245 Bia{\l}ystok\\Poland},
  SUMMARY = {This article is the final step of our attempt to formally 
  construct a prime representing polynomial with 10 variables proposed 
  by Yuri Matiyasevich in \cite{YuriMatiyasevich81}.
\par The first part of the article includes many auxiliary lemmas related 
to multivariate polynomials. We start from the properties of monomials,
among them their evaluation as well as the power function on polynomials 
to define the substitution for multivariate polynomials. For simplicity, 
we assume that a polynomial and substituted ones as i-th variable have 
the same number of variables. Then we study the number of variables that 
are used in given multivariate polynomials. By the used variable we mean 
a variable that is raised at least once to a non-zero power. We consider 
both adding unused variables and eliminating them.
\par The second part of the paper deals with the construction 
of the polynomial proposed by Yuri Matiyasevich. 
First, we introduce a diophantine polynomial over 4 variables
that has roots in integers if and only if indicated variable 
is the square of a natural number, and another two is the square 
of an odd natural number. We modify the polynomial by adding two variables 
in such a way that the root additionally requires the divisibility of these
added variables. Then we modify again the polynomial by adding two variables
to also guarantee the non-negativity condition of one of these variables. 
Finally, we combine the prime diophantine representation proved in 
\cite{HILB10_8.abs} with the obtained polynomial constructing a prime 
representing polynomial with 10 variables.
}}

@article{YuriMatiyasevich81,
      title={Primes are nonnegative values of a polynomial in 10 variables},
      author={Yuri Matiyasevich},
      year={1981},
    JOURNAL = {Journal of Soviet Mathematics},
      VOLUME = {15},
    PAGES = {33--44},
    DOI = {10.1007/BF01404106}
}
