Problem 6

Let

n

be a positive integer. Consider

S=\{(x,y,z) : x,y,z\in\{0,1,\ldots,n\},\ x+y+z>0\}

as a set of

(n+1)^3-1

points in three-dimensional space. Determine the smallest possible number of planes, the union of which contains

S

but does not include

(0,0,0)

Step 4 of 5: Apply the induction hypothesis to Q and conclude m >= 3n

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In plain words

Because $x - n_1 \ne 0$ on the smaller grid, $Q$ inherits the exact vanishing pattern of $P$ on the smaller box, so the induction hypothesis gives the degree bound.

\deg Q \ge (n_1 - 1) + n_2 + n_3 \implies \deg P \ge \deg Q + 1 \ge n_1 + n_2 + n_3 \implies m \ge 3n

Detailed analysis

From Step 3, $Q$ vanishes on $\big(\{0,\ldots,n_1-1\}\times S_2\times S_3\big)\setminus\{(0,0,0)\}$ and $Q(0,0,0) = \dfrac{P(0,0,0)}{-n_1} \ne 0$ . By the induction hypothesis, $\deg Q \ge (n_1-1)+n_2+n_3$ , and since $\deg P \ge \deg((x-n_1)Q) = \deg Q + 1$ , we get $\deg P \ge n_1+n_2+n_3$ , completing the induction. Setting $n_1=n_2=n_3=n$ gives $m = \deg P \ge 3n$ .