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$$1-y^3-x^2+y^3x^8+y^4x^2-y^4x^8$$ is divisible by all three brackets, that is seen from three couplings of terms: $(1-y^3)+(y^4x^2-x^2)+(y^3x^8-y^4x^8)$ is divisible by $y-1$, $(1-x^2)+(y^4x^2-y^4x^...

A simpler answer is given by $x^4y^2 - x^4y - x^2y^2 + x^2 + y - 1=(x+1)f$, where $f$ is the generator of the ideal.

Your general question for $n=1$ (univariate polynomials) is the subject of "Computing sparse multiples of polynomials" by Mark Giesbrecht, Daniel S. Roche, Hrushikesh Tilak: We consider the problem ...

I'll start by answering the question about polynomials with $N$ non-zero coefficients and a non-zero root of multiplicity $N-1$. Polynomials of the form $(x-a)^{N-1}$ are not the only such polynomials!...