The Cullis' determinant of rectangular matrix X equals the Pfaffian of a matrix obtained from X by multiplication and transposition, enabling an efficient polynomial-time algorithm.
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All maps phi and psi satisfying the polynomial preservation equation P(x + λ y) = P(phi(x) + λ psi(y)) are explicitly described using the gradient span L_P for homogeneous P over fields of characteristic zero (and under extra conditions in positive characteristic).
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The Cullis' determinant as Pfaffian
The Cullis' determinant of rectangular matrix X equals the Pfaffian of a matrix obtained from X by multiplication and transposition, enabling an efficient polynomial-time algorithm.
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Nonlinear maps preserving the polynomial
All maps phi and psi satisfying the polynomial preservation equation P(x + λ y) = P(phi(x) + λ psi(y)) are explicitly described using the gradient span L_P for homogeneous P over fields of characteristic zero (and under extra conditions in positive characteristic).