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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 which is referred to as an unmixed polynomial system By Kushnirenko s principle we should expect a lower bound if not all monomials appear in every polynomial 3 2000 Mathematics Subject Classification 14P99 Key words and phrases fewnomials sparse polynomial systems Sottile supported by NSF grant DMS 0701050 and Texas A M ITRAG 1 2 FRE DE RIC BIHAN AND FRANK SOTTILE Such an approach to fewnomial bounds where we take into account differing structures of the polynomials was in fact the source of the first result in this subject In 1978 Sevostyanov proved there is a function N d m such that if the polynomial f x y has degree d and the polynomial g x y has m terms then the system 4 f x y g x y 0 has at most N d m non degenerate positive solutions This result has unfortunately never been published A special case was recently refined by Avendan o 1 who showed that if f is linear then 4 has at most 6m 4 real solutions Li Rojas and Wang 10 showed that a fewnomial system 4 where f has 3 terms will have at most 2m 2 positive solutions when m 3 the bound is lowered to 5 More generally they showed that the number of positive solutions to a system 5 g1 x1 xn g2 x1 xn gn x1 xn 0 is at most n n2 nm 1 when each of g1 gn 1 is a trinomial and gn has m terms These bounds are significantly smaller than the corresponding bounds of 5 which m 1 2 are e 4 3 2 2 nm 1 in both cases Their methods require that at most one polynomial is not a trinomial and apparently do not generalize However their results show that the fewnomial bound can be improved when the polynomials have additional structure We take the first steps towards improving the fewnomial bounds 2 and 3 when the polynomials have additional structure but no limit on their numbers of monomials That is if the polynomial gi in 5 has 2 li terms with li 0 we seek bounds on the number l of non degenerate positive solutions that are smaller in order than 2 2 nl where l n 1 is the total number of terms in all polynomials Note that l l1 ln The reason for our choice of parameterization of these systems is that if some li 0 there is a change of variables which reduces the number of variables eliminates gi from the list polynomials and does not change the number of monomials in the other polynomials nor the number of positive solutions A zero of a system of polynomial equation is non degenerate if the differentials of the equations at that zero are linearly independent Theorem 1 Suppose that each polynomial gi in 5 has a constant term but otherwise all monomials are distinct so that the system involves l n 1 monomials where l l1 ln If the sublattice of Zn spanned by differences of exponent vectors has odd index in Zn then the number of non degenerate non zero real solutions 5 is at most e4 3 2l l 2 4 l1 ln and the number of those which are positive is at most e2 3 2l l 2 4 l1 ln The bounds of Theorem 1 are strictly smaller than those of 2 5 for X l nl l1 ln the sum over all 0 li with l1 ln l In Theorem 1 the bound for positive solutions holds with no assumption on the index of the sublattice spanned by differences of exponent vectors and holds even if we allow A description of this and much more is found in Anatoli Kushnirenko s letter to Sottile 9 FEWNOMIAL BOUNDS FOR COMPLETELY MIXED POLYNOMIAL SYSTEMS 3 real number exponents We only need to prove this for n 2 as these bounds exceed Descartes bound when n 1 We establish Theorem 1 by modifying the arguments of 2 5 In particular we apply a version of Gale duality 6 to replace the system of polynomials by a system of master functions in the complement of a hyperplane arrangement in Rl and then estimate the number of solutions by repeated applications of the Khovanskii Rolle Theorem applied to successive Jacobians of the system of master functions This modification is not as straightforward as we have just made it sound First the arguments we modify require that the hyperplane arrangement be in general position in RPl but in the case here the hyperplanes are arrangements of certain normal crossings divisors in the product of projective spaces RPl1 RPln We exploit the special structure of chambers in this complement together with the multihomogenity of the Jacobians to obtain the smaller bounds of Theorem 1 A more fundamental yet very subtle modification in the arguments is that they require certain successive Jacobians to meet transversally While this can be arranged in 2 5 by varying the parameters we do not have such …


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