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## 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

*i*th polynomial has

*m*_{i}
monomials has at most

(*m*_{1}-1)(*m*_{2}-1)...(*m*_{n}-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

*x*^{108} + 1.1*y*^{54}
- 1.1*y* = *y*^{108}
+ 1.1*x*^{54} - 1.1*x* = 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

*f*_{1}(*x*,*y*) =
*f*_{2}(*x*,*y*) = 0 ,

where

*f*_{1} has 3 terms and

*f*_{2}
has

*m* terms, and every solution is
simple, has at most

2 ( 2^{m-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.

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Systems

** Previous:** 2.iii.b. Fewnomial Bounds