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In modern number theory, it is becoming increasingly common to make computer calculations using p-adic numbers. Just as for real numbers, this involves working with finite approximations and managing the resulting errors as they propagate through the computation. In general, the most flexible framework for this seems to be the natural p-adic analogue of floating-point arithmetic.
David Robbins discovered an example of a computation in which errors in p-adic arithmetic do not appear to compound as is typical: the Dodgson (Lewis Carroll) condensation recurrence for computing determinants.
Based on numerical evidence, he conjectured that the loss of p-adic accuracy during a p-adic floating point computation of the condensation recurrence is controlled by the maximum valuation of any denominator appearing in the computation (i.e., by the largest precision loss at a single step rather than the sum of these).
Additional numerical evidence suggests that this conjecture should follow from a purely algebraic statement applicable to arbitrary cluster algebras.
Using a power series deformation of the caterpillar lemma, we prove a weaker algebraic statement which implies a direct analogue of the Robbins conjecture for some simpler recurrences (Somos-4, Somos-5, Markoff). For a recurrence derived from a general cluster algebra, we obtain a weaker analogue of the Robbins conjecture in which the bound is multiplied by a small positive integer depending on the recurrence (e.g., 3 for condensation).
Joint work with Joe Buhler (CCR La Jolla).No Notes/Supplements Uploaded No Videos Uploaded