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Heights of varieties in multiprojective spaces and arithmetic Nullstellensätze

Carlos D’Andrea, Teresa Krick, Martín Sombra (2013)

Annales scientifiques de l'École Normale Supérieure

We present bounds for the degree and the height of the polynomials arising in some problems in effective algebraic geometry including the implicitization of rational maps and the effective Nullstellensatz over a variety. Our treatment is based on arithmetic intersection theory in products of projective spaces and extends to the arithmetic setting constructions and results due to Jelonek. A key role is played by the notion of canonical mixed height of a multiprojective variety. We study this notion...

Solving linear systems of equations over integers with Gröbner bases

Amir Hashemi (2014)

Acta Arithmetica

We introduce a novel application of Gröbner bases to solve (non-homogeneous) systems of integer linear equations over integers. For this purpose, we present a new algorithm which ascertains whether a linear system of equations has an integer solution or not; in the affirmative case, the general integer solution of the system is determined.

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