## On values of a polynomial at arithmetic progressions with equal products.(English)Zbl 0837.11015

Let $$f \in \mathbb{Q} [X]$$ be monic of degree $$\nu$$. Further, let $$d_1, d_2, \ell,m$$ with $$\ell < m$$ and $$\text{gcd} (\ell, m)=1$$ be given positive integers. The authors consider the equation $f(x) f(x+d_1) \dots f \bigl( x+(\ell k - 1) d_1 \bigr)=f(y) f(y+d_2) \dots f \bigl( y+(mk - 1) d_2 \bigr) \tag{1}$ in integers $$x,y$$ and $$k \geq 2$$ for which $$f(x+jd_1) \neq 0$$ $$(0 \leq j \leq \ell k - 1)$$. In [Acta Arith. 68, 89-100 (1994; Zbl 0812.11023)] the authors considered the case $$f(X)=X$$. The results obtained in this paper are extended in the present paper to the following:
Theorem. Under the conditions given above, equation (1) implies that $$k$$ is bounded by an effectively computable (e.c.) number depending only on $$d_1, d_2, m$$, and $$f$$. Further, let $$f$$ be a power of an irreducible polynomial. Then there exists an e.c. number $$C_3=C_3 (d_1, d_2, m,f)$$ such that (1) implies that $\max \bigl\{ |x |, |y |, k \bigr\} \leq C_3$
unless $$\ell=1$$, $$m=k=2$$, $$d_1=2d^2_2$$, $$f(X)=(X+r)^\nu$$ with $$r \in \mathbb{Z}$$, and $$x+r=(y+r) (y+r+3d_2)$$.

### MSC:

 11D61 Exponential Diophantine equations

Zbl 0812.11023
Full Text: