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FEWNOMIAL BOUNDS FOR COMPLETELY MIXED POLYNOMIAL SYSTEMS FRE DE RIC BIHAN AND FRANK SOTTILE Abstract We give a bound for the number of real solutions to systems of n polynomials in n variables where the monomials appearing in different polynomials are distinct This bound is smaller than the fewnomial bound if this structure of the polynomials is not taken into account Introduction In 1980 A Khovanskii 8 showed that a system of n polynomials in n variables involving l n 1 distinct monomials has less than l n 1 2 2 n 1 l n non degenerate positive solutions This fundamental result established the principle that the number of real solutions to such a system should have an upper bound that depends only upon its number of terms Such results go back to Descartes 7 whose rule of signs implies that a univariate polynomial having l 1 terms has at most l positive zeroes This principle was formulated by Kushnirenko who coined the term fewnomial that has come to describe results of this type Khovanskii s bound 1 is the specialization to polynomials of his bound for a more general class of functions Recently the significantly lower bound of e2 3 2l l 2 2 n 4 was shown 5 for polynomial fewnomial systems This took advantage of some geometry specific to polynomial systems but was otherwise based on Khovanskii s methods The significance of this bound is that it is sharp in the sense that for fixed l there are systems with O nl positive solutions 4 Modifying the proof 2 leads to the bound e4 3 2l l 2 n 4 for the number of real solutions when the differences of exponent vectors of the monomials generate the integer lattice this condition disallows trivial solutions that differ from other solutions only by some predictable signs Otherwise if Zn is generated by the differences of the exponents then 3 is a bound for the number of real zeroes divided by the order of the 2 torsion subgroup in Zn These bounds hold in particular if each of the polynomials involve the same 1 l n monomials



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