Geodesic
The complexity of additive computations of the sets of integer linear forms
Zapiski Nauchnykh Seminarov POMI, Theoretical application of methods of mathematical logic. Part III, Tome 105 (1981), pp. 53-61 Cet article a éte moissonné depuis la source Math-Net.Ru

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An additive computation of a set of linear forms may be presented as the consequence of square matrices $Q_1,\dots,Q_T$ ($Q_i$ equals the unit matrix increased or decreased by 1 in some entry). Thus the additive complexity of a set is the length of the corresponding shortest consequence. A connection between the additive complexity of a set with coefficient matrix $A$ and the complexity of a set with matrix $A^T$ is proved. 
@article{ZNSL_1981_105_a6,
     author = {A. F. Sidorenko},
     title = {The complexity of additive computations of the sets of integer linear forms},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {53--61},
     year = {1981},
     volume = {105},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1981_105_a6/}
}
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A. F. Sidorenko. The complexity of additive computations of the sets of integer linear forms. Zapiski Nauchnykh Seminarov POMI, Theoretical application of methods of mathematical logic. Part III, Tome 105 (1981), pp. 53-61. http://geodesic.mathdoc.fr/item/ZNSL_1981_105_a6/