12 1. OVERVIEW Example 1.16. We close this overview with one example from this theory. Let t, x, y, z be indeterminates, and consider a sparse polynomial of the form (1.11) c4 txyz + c3(txz + xyz) + c2(tx + xz + yz) + c1(x + z) + c0 , where the coeﬃcients c0,...,c4 are real numbers. Theorem 1.17. A system involving four polynomials of the form (1.11) has six solutions, at least two of which are real. We make some remarks to illustrate the ingredients of this theory. First, the monomials in the sparse system (1.11) are the integer points in the order polytope of the poset P , P := t y x z . That is, each monomial corresponds to an order ideal of P (a subset which is closed upwards). The number of complex roots is the number of linear extensions of the poset P . There are six, as each is a permutation of the word txyz where t precedes x and y precedes z. One result (ii) characterizes polytopes whose associated polynomial systems will have a lower bound, and many order polytopes satisfy these conditions. An- other result (iv) computes that lower bound for certain families of polynomials with support an order polytope. Polynomials in these families have the form (1.11) in that monomials with the same total degree have the same coeﬃcient. For such polynomials, the lower bound is the absolute value of the sum of the signs of the permutations underlying the linear extensions. We list these for P . permutation txyz tyxz ytxz tyzx ytzx yztx sum sign + − + + − + 2 This shows that the lower bound in Theorem 1.17 is two. Table 1.1 records the frequency of the different numbers of real solutions in each of 10,000,000 instances of this polynomial system, where the coeﬃcients were chosen uniformly from [−200, 200]. This computation took 13 gigahertz-hours. Table 1.1. Observed frequencies of numbers of real solutions. number of real solutions 0 2 4 6 frequency 0 9519429 0 480571 The apparent gap in the numbers of real solutions in Table (1.1) (four does not seem a possible number of real solutions) is proven for the system of Example 1.16 in Section 8.3. This is the first instance we have seen of this phenomena of gaps in the numbers of real solutions. More are found in [138], [123], and some are presented in Chapters 8, 13, and 14. Many examples of lower bounds continue to be found, e.g. [3].

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