Problem 6

An ordered pair

(x,y)

of integers is called a primitive point if

\gcd(x,y)=1

. Given a finite set

S

of primitive points, prove that there exist a positive integer

n

and integers

a_0,a_1,\ldots,a_n

such that for every

(x,y)\in S

, we have

a_0x^n+a_1x^{n-1}y+a_2x^{n-2}y^2+\cdots+a_{n-1}xy^{n-1}+a_ny^n=1

Step 6 of 8: Build the corrected polynomial

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In plain words

The first term preserves old values, while the product correction vanishes there.

Q(x,y)=P(x,y)^U+C(ax+by)^V\prod_{i=1}^{k-1}(y_ix-x_iy)

Detailed analysis

Let $C=(1-P(x_k,y_k)^U)/D$ , which is an integer because $P(x_k,y_k)^U\equiv1\pmod{|D|}$ . Define $Q(x,y)=P(x,y)^U+C(ax+by)^V\prod_{i=1}^{k-1}(y_ix-x_iy)$ . Both summands are homogeneous of degree nU, so Q has integer coefficients and positive degree.