Any polynomial which is hard to count but easy to decide? Every monotone arithmetic circuit, i.e. a $\{+,\times\}$-circuit, computes some multivariate
polynomial $F(x_1,\ldots,x_n)$ with nonnegative integer coefficients. Given a polynomial
$f(x_1,\ldots,x_n)$, the circuit


 computes $f$ if $F(a)=f(a)$ holds for all $a\in \mathbb{N}^n$;
 counts $f$ if $F(a)=f(a)$ holds for all $a\in\{0,1\}^n$;
 decides $f$ if $F(a)&gt;0$ exactly when $f(a)&gt;0$ holds for all  $a\in\{0,1\}^n$.
 

I know explicit polynomials $f$ (even multilinear) showing that the circuit-size gap "computes/counts" can be exponential. My question concerns the gap "counts/decides".


Question 1: Does anybody know of any polynomial $f$ which is exponentially harder to count than to decide by $\{+,\times\}$-circuits?


As a possible candidate, one could take the PATH polynomial whose variables correspond to edges of the complete graph $K_n$ on $\{1,\ldots,n\}$, and each monomial corresponds to a simple path from node $1$ to node $n$ in $K_n$.
This polynomial can be decided by a circuit of size $O(n^3)$ implementing, say, the 
Bellman-Ford dynamic programming algorithm, and it is relatively easy to show that every $\{+,\times\}$-circuit computing PATH must have size $2^{\Omega(n)}$.
 
 On the other hand, every circuit counting
PATH solves the $\#$PATH problem, i.e. counts the number of $1$-to-$n$ paths in the specified by the corresponding $0$-$1$ input subgraph of $K_n$. This is a so-called $\#$P-complete problem. So, we all "believe" that PATH cannot have any counting $\{+,\times\}$-circuits of polynomial size. The "only" problem is to prove this ...

I can show that every $\{+,\times\}$-circuit counting a related  Hamiltonian path polynomial HP requires exponential size. Monomials of this polynomial correspond to $1$-to-$n$ paths in $K_n$ containing
all nodes. 
Unfortunately, the reduction of $\#$HP to $\#$PATH by Valiant requires to compute the inverse of the Vandermonde matrix, and hence cannot be implemented by a $\{+,\times\}$-circuit.


Question 2: Has anybody seen a monotone reduction of $\#$HP to $\#$PATH?


And finally: 


Question 3: Was a "monotone version" of the class $\#$P considered at all? 


N.B. Note that I am talking about a very restricted class of circuits: monotone arithmetic circuits! In the class of $\{+,-,\times\}$-circuits, Question 1 would be just unfair to ask at all:  no lower bounds larger than $\Omega(n\log n)$ for such circuits, even when required to compute a given polynomial on all inputs in $\mathbb{R}^n$, are known.
Also, in the class of such circuits, a "structural analogue" of Question 1 --
are there $\#$P-complete polynomials which can be decided by poly-size $\{+,-,\times\}$-circuits? --
has an affirmative answer. Such is, for example, the permanent polynomial PER$=\sum_{h\in S_n}\prod_{i=1}^n x_{i,h(i)}$. 

ADDED:  Tsuyoshi Ito answered Question 1 with a very simple trick. Still, Questions 2 and 3 remain open. The counting status of PATH is interesting in its own both because it is a standard DP problem and because it is #P-complete.
(I am posting my comments as an answer in response to the OP’s request.)

As for Question 1, let fn: {0,1}n→ℕ be a family of functions whose arithmetic circuit requires exponential size.  Then so does fn+1, but fn+1 is easy to decide by a trivial monotone arithmetic circuit.  If you prefer to avoid constants in monotone arithmetic circuits, then let fn: {0,1}n→ℕ be a family of functions such that the arithmetic circuit for fn requires exponential size and fn(0, …, 0)=0, and consider fn+x1+…+xn.