Next: 3. Enumerative Real Algebraic Geometry
Up: 2.iii. Real Solutions to Sparse Polynomial Systems
Previous: 2.iii.b. Fewnomial Bounds

## 2.iii.c. Kushnirenko's Conjecture

It may be more feasible to look for bounds on r+(A), the number of solutions with positive coordinates. Kushnirenko made§ the following conjecture.

Conjecture 2.7 (Kushnirenko (see [BCS], p. 300) A system of n polynomials in n variables with only simple solutions whose ith polynomial has mi monomials has at most

(m1-1)(m2-1)...(mn-1)

solutions with positive coordinates.

For a system of two trinomials in 2 variables, Kushnirenko's conjecture asserts that r+ is at most 4. In 2000, Haas [Haa] gave the counterexample

x108 + 1.1y54 - 1.1y   =   y108 + 1.1x54 - 1.1x   =   0 ,

a system of two trinomials with 5 solutions having positive coordinates. Although Kushnirenko's conjecture is false, the question remains: Is the true value for r+(A) (or r(A)) closer to Khovanskii's bound or to the number in Kushnirenko's conjecture? Recent work suggests that it is the latter.

Theorem 2.8 (Li, Rojas, and Wang [LRW])   A system of 2 polynomials in 2 variables

f1(x,y)   =   f2(x,y)   =   0 ,

where f1 has 3 terms and f2 has m terms, and every solution is simple, has at most

2 ( 2m-1 -1 )

solutions with positive coordinates. When m=3, this bound may be improved to 5.

When m = 3, so we have two trinomials in 2 variables, this gives a bound of 5, so Theorem 2.8 give implies that Haas's construction with 5 solutions is the best possible. What if m=4? Is there a system consisting of one real trinomial and one tetranomial with 14 positive roots?

Remark 2.9   Khovanskii's bound for the number of real solutions with positive coordinates holds also for systems of power functions, where the exponents of monomials are arbitrary real numbers. One might suppose that this added generality is the source of the large size of his bound and its apparent lack of tightness for polynomial systems. However, Napoletani [Na] has shown that the complexity bounds are the same for both polynomials and for power functions.

§Apparently, Kushnirenko did not believe this bound. Nonetheless, the conjecture and its attribution have passed into folklore.

Next: 3. Enumerative Real Algebraic Geometry
Up: 2.iii. Real Solutions to Sparse Polynomial Systems
Previous: 2.iii.b. Fewnomial Bounds