A two-variable polynomial eigenvalue equation: why only degree two survives

stardew_afterdark·Aug 21, 2026

Trying individual quadratic monomials finds examples, but how do we rule out higher-degree terms without assuming the answer is quadratic? Problem: Find all polynomial functions

P(x, y)

with real coefficients which for all real numbers

x

and

y

satisfy

P(x + y, x - y) = 2P(x, y).

Source: Baltic Way 2020 · MNET-01GH

Open MNET-01GH in the question bank →

3 replies

shieldonbreak·Aug 21, 2026

Apply the coordinate transformation twice. It sends (x,y) to (2x,2y), so

P(2x,2y)=4P(x,y)

. Comparing coefficients forces every nonzero monomial to have total degree 2.

stardew_afterdark·Aug 21, 2026

Then write

P=cx^2+axy+by^2

. Substituting into the original equation gives c=a+b, and the other coefficient equations are the same condition.

shieldonbreak·Aug 21, 2026

Thus

P(x,y)=(a+b)x^2+axy+by^2

for arbitrary real a,b, including the zero polynomial. Each satisfies the original equation directly. The doubled transformation gave a necessary degree restriction; the final substitution imposes the remaining condition.

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A two-variable polynomial eigenvalue equation: why only degree two survives

stardew_afterdark·Aug 21, 2026

Trying individual quadratic monomials finds examples, but how do we rule out higher-degree terms without assuming the answer is quadratic? Problem: Find all polynomial functions

P(x, y)

with real coefficients which for all real numbers

x

and

y

satisfy

P(x + y, x - y) = 2P(x, y).

Source: Baltic Way 2020 · MNET-01GH

Open MNET-01GH in the question bank →

3 replies

shieldonbreak·Aug 21, 2026

Apply the coordinate transformation twice. It sends (x,y) to (2x,2y), so

P(2x,2y)=4P(x,y)

. Comparing coefficients forces every nonzero monomial to have total degree 2.

stardew_afterdark·Aug 21, 2026

Then write

P=cx^2+axy+by^2

. Substituting into the original equation gives c=a+b, and the other coefficient equations are the same condition.

shieldonbreak·Aug 21, 2026

Thus

P(x,y)=(a+b)x^2+axy+by^2