Artinian Gorenstein algebras of embedding dimension four and socle degree three over an arbitrary field

Let k be an arbitrary field, A be a standard graded Artinian Gorenstein k-algebra of embedding dimension four and socle degree three, and pi from P to A be a surjective graded homomorphism from a polynomial ring with four variables over k onto A. We give the minimal generators of the kernel of pi and the minimal homogeneous resolution of A by free P-modules. We give formulas for the entries in the matrices in the resolution in terms of the coefficients of the Macaulay inverse system for A. We have implemented these formulas in Macaulay2 scripts. The kernel of pi has either 6, 7, or 9 minimal generators. The number of minimal generators and the precise form of the minimal resolution are determined by the rank of a 3 by 3 symmetric matrix of constants that we call SM. If the kernel of pi requires more than six generators, then we prove that the kernel of pi is the sum of two linked perfect ideals of grade three. If the the kernel of pi is six-generated, then we prove that A is a hypersurface section of a codimension three Gorenstein algebra. Our approach is based on the structure of Gorenstein-linear resolutions and the theorem that, except for exactly one exception, A has the weak Lefschetz property.