@ARTICLE{HILB10_7.MIZ,
  AUTHOR = {P\k{a}k, Karol},
  TITLE = {Prime Representing Polynomial with 10 Unknowns -- Introduction},
  SECTION1 = {Preliminaries},
  SECTION2 = {Selected Operations on Set Families},
  SECTION3 = {Function where each Value is Repeated an Even Number of Times},
  SECTION4 = {Cartesian Product of Domains in Finite Sequences},
  SECTION5 = {Some Operations on Finite Sequences},
  SECTION6 = {Combination of Sign and Characteristic Functions},
  SECTION7 = {Product over all Combinations of Sings},
  DAY = {30},
  MONTH = {September},
  YEAR = {2022},
  ADDRESS1 = {Institute of Computer Science\\University of Bia{\l}ystok\\
        Konstantego Cio{\l}kowskiego 1M, 15-245 Bia{\l}ystok\\Poland},
  SUMMARY = {The main purpose of the article is to construct
  a sophisticated polynomial proposed by  Matiyasevich and Robinson \cite{MR75}
  that is often used to reduce the number of
  unknowns in diophantine representations. The polynomial
  $$J_k(a_1,\ldots,a_k,x) = \prod_{\epsilon_1,\ldots,\epsilon_k\in\{\pm 1\}}(x+\epsilon_1\!\sqrt{a_1}+\epsilon_2\!\sqrt{a_2}W+\ldots+\epsilon_k\!\sqrt{a_k}W^{k\!-\!1})$$
  with $W=\sum_{i=1}^k x_i^2$ has integer coefficients and $J_k(a_1,\ldots,a_k,x)=0$
  for some $a_1,\ldots,a_k,x\in\mathbb{Z}$ if and only if $a_1,\ldots,a_k$ are all squares.
  However although it is nontrivial to observe that this expression is a polynomial, i.e.,
  eliminating similar elements in the product of all combination of sings
  we obtain an expression where every square root will occur with an even power.}}

@ARTICLE{MR75,
  AUTHOR =       {Yuri Matiyasevich and Julia Robinson},
  TITLE =        {Reduction of an Arbitrary Diophantine Equation to One in 13 Unknowns},
  JOURNAL =      {Acta Arithmetica},
  YEAR =         {1975},
  volume =       {27},
  pages =        {521--553}
}

@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}
}

@article{sun2021results,
      title={Further results on {H}ilbert's {T}enth {P}roblem},
      author={Zhi-Wei Sun},
      year={2021},
    JOURNAL = {Science China Mathematics},
      VOLUME = {64},
    PAGES = {281--306},
    DOI = {10.1007/s11425-020-1813-5}
}
