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Mathoverflow.net news digest

  • 0 days

    Uniform bound on fiber size of a polynomial map

    This is a follow-up to my previous question.
    In the answer to the above, the following was demonstrated. If $P:\mathbb Z^n\to\mathbb Z^N$, $1<n\leq N$, is a polynomial map such that $DP(x)$ has full rank at every $x\in\mathbb Z^n$, then each fiber $P^{-1}(b)$ of $P$ is finite....

  • 0 days

    Closed form for certain family of determinants

    Let
    $T(n,m,k)$ be the family of integer coefficients such that $T(n,m,k) = \det(M_n)$ where $M_n$ is the $n \times n$ matrix with $(i,j)$-th element ${i+m \brace j+k}$.
    $R(n,m,k)$ be the family of integer coefficients defined for $1 \leqslant k \leqslant \binom{n+m-1}{m}$ such that $R(n,m,k)$ is the product of the elements of the $k$-th composition of $\{ 1,2,\dotsc,n+m-1 \}$ of the size $m$ with elements $e_i$ such...

  • 0 days

    $\aleph_1$-filtered colimits of free abelian groups

    Does the category $\mathbf{FreeAb}$ of free abelian groups have $\aleph_1$-filtered colimits?
    See the nLab for the definition of a $\kappa$-filtered colimit. Here are some thoughts and related observations on this question.
    By Does the category of free abelian groups have sequential colimits? it does not have filtered colimits. But having $\aleph_1$-filtered colimits is a strictly weaker property....

  • 0 days

    Existence of a regular $n$-gon with vertices on $n$ circles sharing a common poi...

    It is well-known that there exist infinitely many equilateral triangles whose vertices lie on the edges of a given triangle, and infinitely many squares whose vertices lie on the edges of a given quadrilateral. Motivated by these observations, I would...

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