Finding a nonzero polynomial involving the floor function

Question

Find a nonzero polynomial $P(x,y)$ where the coefficients are integers such that $P(\lfloor a \rfloor, \lfloor 2a \rfloor) = 0$ $\forall a \in$ $\mathbb{R}$

Answer 1

Hint: $\lfloor a \rfloor = a - f$ where $f$ is the fractional part. Then, there are three possibilities:

1. $f < 0.5$, giving $\lfloor 2a \rfloor = 2a - 2f$
2. $f = 0.5$, giving $\lfloor 2a \rfloor = 2a$
3. $f > 0.5$, giving $\lfloor 2a \rfloor = 2a - 2f + 1$

These are easily verified. But case 2 seems problematic in our problem. Can you eliminate case 2 (hint: what is $2f$ in case 2)?

Answer 2

Hint: $\,2 \lfloor a \rfloor -1 \le 2a-1 \lt \lfloor 2a \rfloor \le 2a \lt 2 \left(\lfloor a \rfloor +1\right) = 2 \lfloor a \rfloor + 2\,$, so $2 \lfloor a \rfloor\le \lfloor 2a \rfloor \lt 2 \lfloor a \rfloor + 2\,$. It follows that $\,\lfloor 2a \rfloor\,$ can only take one of the two integer values $\,2 \lfloor a \rfloor\,$ or $\,2 \lfloor a \rfloor+1\,$, and therefore:

$$\big(\lfloor 2a \rfloor - 2 \lfloor a \rfloor \big)\big(\lfloor 2a \rfloor - 2 \lfloor a \rfloor - 1\big) = 0 \quad\quad\forall a \in \mathbb{R}$$