Progress on the unfair 0-1-polynomials conjecture using linear recurrences and numerical analysis

If the product of two monic polynomials with real nonnegative coefficients has all coefficients equal to 0 or 1, does it follow that all the coefficients of the two factors are also equal to 0 or 1? Here is an equivalent formulation of this intriguing problem: is it possible to weigh unfairly a pair of dice so that the probabilities of every possible outcome (roll them and take the sum) were the same? If the two dice have six faces numbered 1 to 6, it is easy to show that the answer is no. But for general dice with finitely many faces, this is an open problem with no significant advancement since 1937. In this paper we examine, in some sense, the first infinite family of cases that cannot be treated with classical methods: the first die has three faces numbered with 0,2 and 5, while the second die is arbitrary. In other words, we examine factorizations of 0-1-polynomials with one factor equal to $x^5+a x^2 +1$ for some nonnegative $a$. We discover that this case may be solved (that is, necessarily $a=0$ or $a=1$) using the theory of linear recurrence sequences, computation of resultants, a fair amount of analytic and numerical approximations... and a little bit of luck.Comment: 46 pages, 3 figures. Comments, feedback and ideas welcome

Instrumented crutches for gait parameters evaluation

Most of the prototypes of instrumented crutches available in the literature require external motion capture devices to perform a gait analysis and to report the load applied on the crutches with respect to the gait cycle. Motion capture systems with markers require a controlled laboratory with cameras, instead IMU-based systems are more transportable, but the user must be instrumented. A new version of instrumented crutches, previously developed by the authors, allows one to measure the axial forces and to detect the gait phases during two-point assisted walking thanks to the cameras mounted on the lower part of the crutches

Heights of multiprojective cycles and small value estimates in dimension two.

In the first part we recall the theory of multiprojective elimination initiated by P.Philippon and developed by G.Rémond. In particular, we define the eliminant ideal, the resultant forms and the Hilbert-Samuel polynomial for multigraded modules. We then look at subvarieties and cycles of a product of projective spaces, over a number field, and we define their mixed degrees and mixed heights, which measure respectively their geometric and arithmetic complexity. Finally, we define the heights of multiprojective cycles relative to some sets of polynomials, generalizing a previous notion of height due to M.Laurent and D.Roy, and we give detailed proofs for their properties. In the second part we prove that if we have a sequence of polynomials with bounded degrees and bounded integer coefficients taking small values at a pair (a,b) together with their first derivatives, then both a and b need to be algebraic. The main ingredients of the proof include a translation of the problem in multihomogeneous setting, an interpolation result, the construction of a 0-dimensional variety with small height, a result for the multiplicity of resultant forms, and a final descent. This work is motivated by an arithmetic statement equivalent to Schanuel's conjecture, due to D.Roy