Range of unknowns

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< Matiyasevich theorem
Yuri Vladimirovich Matiyasevich (2008), Scholarpedia, 3(7):7095. revision #42336 [link to/cite this article]

Curator: Dr. Yuri Vladimirovich Matiyasevich, Steklov Institute of Mathematics (POMI), St.Petersburg, RUSSIA

Besides exhibiting a Diophantine equation itself, one has to specify what are the admissible values of the unknowns. The same equation may define different sets depending on whether the unknowns range over all integers or only over non-negative integers. However, such restriction doesn't effect the scope of the whole class of Diophantine sets. Indeed, the set defined by a Diophantine equation

(1)
P(a,x_1,\dots,x_m)=0

with integer-valued unknowns is also defined by the Diophantine equation

P(a,y_1-z_1,\dots,y_m-z_m)=0

with unknowns restricted to non-negative integers. Similar, the set defined by the same equation (1) but with unknowns restricted to non-negative integers is also defined by the equation

P(a,p_1^2+q_1^2+r_1^2+s_1^2,\dots,p_m^2+q_m^2+r_m^2+s_m^2)=0

with arbitrary integer values for the unknowns (thanks to Lagrange's four-square theorem stating that every non-negative integer is the sum of four squares).


Suggested by: Mr. Nicolau Leal Werneck, USP, São Paulo, Brazil
Invited by: Dr. Eugene M. Izhikevich, Editor-in-Chief of Scholarpedia, the peer-reviewed open-access encyclopedia
Action editor: Dr. Eugene M. Izhikevich, Editor-in-Chief of Scholarpedia, the peer-reviewed open-access encyclopedia
Reviewer A: Dr. Alexandra Shlapentokh, Department of Mathematics, East Carolina University, Greenville, NC
Reviewer B: Dr. Martin Davis, New York University, NY
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