Kaplanski's Zero Divisor Conjecture envisions that for a torsion-free group Gand an integral domain R, the group ring R[G] does not contain non-trivial zerodivisors. We define the length of an element a in R[G] as the minimalnon-negative integer k for which there are ring elements r_1,...,r_k in R andgroup elements g_1,...,g_k in G such that a = r_1 g_1+...+r_k g_k. Weinvestigate the conjecture when R is the field of rational numbers. By areduction to the finite field with two elements, we show that if ab = 0 fornon-trivial elements in the group ring of a torsion-free group over therationals, then the lengths of a and b cannot be among certain combinations.More precisely, we show for various pairs of integers (i,j) that if one of thelengths is at most i then the other length must exceed j. Using combinatorialarguments we show this for the pairs (3,6) and (4,4). With a computer-assistedapproach we strengthen this to show the statement holds for the pairs (3,16)and (4,7). As part of our method, we describe a combinatorial structure, whichwe call matched rectangles, and show that for these a canonical labeling can becomputed in quadratic time. Each matched rectangle gives rise to a presentationof a group. These associated groups are universal in the sense that there is nocounterexample to the conjecture among them if and only if the conjecture istrue over the rationals.
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