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Heinrich Massold

Publications and source records attributed to Heinrich Massold.

6 recordsLinked to original sources
arXiv2025

Transcendence degrees of fields generated by exponentials of products

Let $θ=(θ_1,\ldots,θ_m) \in \R^m, κ=(κ_1,\ldots,κ_n) \in \R^n$ be two tuples of real numbers each linearly independent over $\Q$, and $T$ the transcendence degree of the field generated by $\{\exp(θ_i κ_j) | i=1,\ldots,m, \; j=1,\ldots,n \}$ over $\Q$. The estimate $T \geq \frac{mn}{m+n} -1$ has been conjectured for some time but could only be proved under additional hypotheses for $θ$ and $κ$. This paper proves a weaker estimate for $T$ while also reducing the strong estimate to a prominent conjecture on intersections of subvarieties of split tori with subgroups.

math.NT↗
arXiv2016

Diophantine Approximation on projective Varieties I: Algebraic distance and metric Bézout Theorem

For two properly intersecting effective cycles in projective space X,Y, and their intersection product Z, the metric Bezout Theorem relates the degrees, heights of X,Y, and Z, as well as their distances and algebraic distances to a given point theta. Applications of this Theorem are in the area of Diophantine Approximation, giving estimates for approximation properties of Z with respect to $θ$ against the ones of X, and Y.

math.NT↗
arXiv2010

Diophantine Approximation on varieties V: Algebraic independence criteria

For a tuple $(θ_1,..,θ_M)$ of complex number, buliding on the approximation techniques in earlier papers of this series, this paper engages in deducing lower estimates on the transcendence degree of the field generated by $θ_1, ..., θ_M$ over the field of rational numbers from the approximability of the point $θ=(1,θ_1,...,θ_M)$ in projective space by hypersurfaces. The first given result is an new proof of an algebraic independence criterion, that was formerly proved by Laurent and Roy, and generalizes the Philippon criterion by introducing also evaluations of derivatives of global sections. The second result is a new kind of algebraic independence criteria that has a wider range of applicabilty than the first one.

math.NT↗
arXiv2009

Diophantine Approximation on Varieties IV: Derivated algebraic distance and derivative metric Bezout Theorem

The metric Bezout Theorem proved in an earlier paper can be extended to a derivative version that compares derivatives of the algebraic distance of a point $θ$ to two properly intersecting cycles in projective space with the derivatives of the algebraic distance of $θ$ to their intersection. This improvement can be used to make algebraic independence criteria, to be proved in a forthcoming paper, more flexible and to refine Approximation results that are proved using the metric Bezout Theorem.

math.AG↗
arXiv2007

Diophantine Approximation on varieties III: Approximation of non-algebraic points by algebraic points

For $θ$ a non-algebraic point on a quasi projective variety over a number field, I prove that $θ$ has an approximation by a series of algebraic points of bounded height and degree which is essentially best possible. Applications of this result will include a proof of a slightly strengthened version of the Philippon criterion, some new algebraic independence criteria, statements concerning metric transcendence theory on varieties of arbitrary dimension, and a rather accurate estimate for the number of algebraic points of bounded height and degree on quasi projective varieties over number fields.

math.NT↗