Diophantine quadruples in Z[i][X]

In this paper, we prove that every Diophantine quadruple in Z [ i ] [ X ] is regular. More precisely, we prove that if { a , b , c , d } is a set of four non-zero polynomials from Z [ i ] [ X ] , not all constant, such that the product of any two of its distinct elements increased by 1 is a square o...

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Bibliographic Details
Published in:Periodica mathematica Hungarica 2021-06, Vol.82 (2), p.198-212
Main Authors: Filipin, Alan, Jurasić, Ana
Format: Article
Language:English
Subjects:
Mathematics
Mathematics and Statistics
Polynomials
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Summary:In this paper, we prove that every Diophantine quadruple in Z [ i ] [ X ] is regular. More precisely, we prove that if { a , b , c , d } is a set of four non-zero polynomials from Z [ i ] [ X ] , not all constant, such that the product of any two of its distinct elements increased by 1 is a square of a polynomial from Z [ i ] [ X ] , then ( a + b - c - d ) 2 = 4 ( a b + 1 ) ( c d + 1 ) .
ISSN:0031-5303
1588-2829
DOI:10.1007/s10998-020-00353-y