Building Extremal Fewnomial Lower Bounds Over Local Fields, and the Adelic Tau Conjecture

Details Building Extremal Fewnomial Lower Bounds Over Local Fields, and the Adelic Tau Conjecture Building Extremal Fewnomial Lower Bounds Over Local Fields, and the Adelic Tau Conjecture

2010-11-18

arxiv.org/abs/1011.4128v3

Mathematics

Algebraic Geometry

16 pages, 4 figures, submitted to refereed conference proceedings. Fixes various typos, expands main proofs, and includes (a)

Scientific paper

Consider a system F of n polynomials in n variables, with a total of n+k distinct exponent vectors, over any local field L. We discuss conjecturally tight upper and lower bounds on the maximal number of non-degenerate roots F can have over L, with all coordinates having fixed sign or fixed first digit, as a function of n and k only. For non-Archimedean L we give the first non-trivial lower bounds in the case $k-1 \leq n$; and for general L we give new explicit extremal systems when k=2 and $n \geq 1$. We also briefly review the background behind such bounds, and their application, including connections to variants of the Shub-Smale $\tau$-Conjecture and the P vs. NP Problem. One of our key tools is the construction of combinatorially constrained tropical varieties with maximally many intersections.

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