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grep rough audit - static analysis tool
v2.8 written by @Wireghoul
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normaliz-3.8.9+ds/PyNormaliz/doc/PyNormaliz_Tutorial.ipynb-188- "\n",
normaliz-3.8.9+ds/PyNormaliz/doc/PyNormaliz_Tutorial.ipynb:189: "For the series, the first vector consists of the coefficients of the numerator polynomial, e.g., `[1,14,15]` is $1+14t+15t^2$. The second vector collects the exponents in the denominator. For example `[1,1,1,1]` represents $(1-t)(1-t)(1-t)(1-t)=(1-t)^4$ and `[1,2]` is $(1-t)(1-t^2)$. The last entry is the possible shift of the Hilbert series.\n",
normaliz-3.8.9+ds/PyNormaliz/doc/PyNormaliz_Tutorial.ipynb-190- "\n",
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normaliz-3.8.9+ds/doc/Normaliz.tex-4000-
normaliz-3.8.9+ds/doc/Normaliz.tex:4001:(2) For the lattice normalized volume we need a lattice $L$ of reference. We assume that $\aff(P)\subset\aff(L)$. (It would be enough to have this inclusion after a parallel translation of $\aff(P)$.) Choosing the origin in $L$, one can assume that $\aff(L)$ is a vector subspace of $\RR^d$ so that we can identify it with $\RR^d$ after changing $d$ if necessary. After a coordinate transformation we can further assume that $L=\ZZ^d$ (in general this is not an orthogonal change of coordinates!). To continue we need that $\aff(P)$ is a rational subspace. There exists $k\in\NN$ such that $k\aff(P)$ contains a lattice simplex. The lattice normalized volume $\vol_L$ of $kP$ is then given by the Lebesgue measure on $k\aff(P)$ in which the smallest possible lattice simplex in $k\aff(P)$ has volume $1$. Finally we set $\vol_L(P)=\vol_L(kP)/k^r$ where $r=\dim(P)$.
normaliz-3.8.9+ds/doc/Normaliz.tex-4002-
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normaliz-3.8.9+ds/doc/Normaliz.tex-4165-\end{Verbatim}
normaliz-3.8.9+ds/doc/Normaliz.tex:4166:After finishing the $49$ ``small'' simplicial cones, Normaliz takes on the $47$ ``large'' simplicial cones, and does them by project-and-lift (including LLL). Therefore one can say that Normaliz takes a hybrid approach to lattice points in primal mode.
normaliz-3.8.9+ds/doc/Normaliz.tex-4167-
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normaliz-3.8.9+ds/doc/Normaliz.tex-4520-$$
normaliz-3.8.9+ds/doc/Normaliz.tex:4521:where $C'=C\setminus \mathcal F$ and $\mathcal F$ is the union of a set of
normaliz-3.8.9+ds/doc/Normaliz.tex-4522-faces
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normaliz-3.8.9+ds/doc/Normaliz.tex-5166- in the triangulation, and the next line contains the
normaliz-3.8.9+ds/doc/Normaliz.tex:5167: number $m+1$ where $m=\rank \EE$. Each of the following
normaliz-3.8.9+ds/doc/Normaliz.tex-5168- lines specifies a simplicial cone $\Delta$: the first
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normaliz-3.8.9+ds/doc/Normaliz.tex-7307-$$
normaliz-3.8.9+ds/doc/Normaliz.tex:7308:where $\lambda$ is an affine form, i.e., a non-constant map $\lambda:\RR^d\to\RR$, $\lambda(x)=\alpha_1x_1+\dots+\alpha_dx_d+\beta$ with $\alpha_1,\dots,\alpha_d,\beta\in\RR$. If $\beta=0$ and $\lambda$ is therefore linear, then the halfspace is called \emph{linear}. The halfspace is \emph{rational} if $\lambda$ is \emph{rational}, i.e., has rational coordinates. If $\lambda$ is rational, we can assume that it is even \emph{integral}, i.e., has integral coordinates, and, moreover, that these are coprime. Then $\lambda$ is uniquely determined by $H_\lambda^+$. Such integral forms are called \emph{primitive}, and the same terminology applies to vectors.
normaliz-3.8.9+ds/doc/Normaliz.tex-7309-
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normaliz-3.8.9+ds/doc/Normaliz.tex-7496-
normaliz-3.8.9+ds/doc/Normaliz.tex:7497:Let $P\subset \RR^d$ be a rational polyhedron and $L\subset \ZZ^d$ be an \emph{affine sublattice}, i.e., a subset $w+L_0$ where $w\in\ZZ^d$ and $L_0\subset \ZZ^d$ is a sublattice. In order to investigate (and compute) $P\cap L$ one again uses homogenization: $P$ is extended to $C(P)$ and $L$ is extended to $\cL=L_0+\ZZ(w,1)$. Then one computes $C(P)\cap \cL$. Via this ``bridge'' one obtains the following inhomogeneous version of Gordan's lemma:
normaliz-3.8.9+ds/doc/Normaliz.tex-7498-
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normaliz-3.8.9+ds/doc/Normaliz.tex-7549-A rational cone $C$ and a grading together define the rational
normaliz-3.8.9+ds/doc/Normaliz.tex:7550:polytope $Q=C\cap A_1$ where $A_1=\{x:\deg x=1\}$. In this
normaliz-3.8.9+ds/doc/Normaliz.tex-7551-sense the Hilbert series is nothing but the Ehrhart series of
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normaliz-3.8.9+ds/doc/Normaliz.tex-7571-$$
normaliz-3.8.9+ds/doc/Normaliz.tex:7572:where $\vol$ is the lattice normalized volume of $Q$ (a lattice simplex of smallest possible volume has volume $1$). The \emph{multiplicity} of $M$, denoted by $e(M)$ is $(r-1)!q_{r-1}=\vol(Q)$.
normaliz-3.8.9+ds/doc/Normaliz.tex-7573-
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normaliz-3.8.9+ds/doc/Normaliz.tex-7596-
normaliz-3.8.9+ds/doc/Normaliz.tex:7597:For the interpretation of the multiplicity $e(N)=\mrank(N)e(M)$ one must first split the module $N$ into a direct sum where each summand bundles the elements whose degrees belong to a fixed residue class modulo $g$. Let $N^0,\dots,N^{g-1}$ be these summands. Then $e(N^k)$ is the dimension normed constant leading coefficient of the Hilbert quasipolynomial of $N^k$ for each $k$, and $e(N)=\sum_k e(N^k)$.
normaliz-3.8.9+ds/doc/Normaliz.tex-7598-
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normaliz-3.8.9+ds/doc/Normaliz.tex-7600-
normaliz-3.8.9+ds/doc/Normaliz.tex:7601:A normal affine monoid $M$ has a well-defined divisor class group. It is naturally isomorphic to the divisor class group of $K[M]$ where $K$ is a field (or any unique factorization domain); see \cite[4.F]{BG}, and especially \cite[4.56]{BG}. The class group classifies the divisorial ideals up to isomorphism. It can be computed from the standard embedding that sends an element $x$ of $\gp(M)$ to the vector $\sigma(x)$ where $\sigma$ is the collection of support forms $\sigma_1,\dots,\sigma_s$ of $M$: $\Cl(M)=\ZZ^s/\sigma(\gp(M))$. Finding this quotient amounts to an application of the Smith normal form to the matrix of $\sigma$.
normaliz-3.8.9+ds/doc/Normaliz.tex-7602-
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normaliz-3.8.9+ds/doc/Normaliz.tex-8724-\end{Verbatim}
normaliz-3.8.9+ds/doc/Normaliz.tex:8725:Here ``Embedding'' refers to $\phi$ and ``Projection'' to $\pi$ (though $\pi$ is not always surjective). The ``Annihilator'' is the number $c$ above. (It annihilates $\ZZ^r$ modulo $\pi(\ZZ^n)$.)
normaliz-3.8.9+ds/doc/Normaliz.tex-8726-
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normaliz-3.8.9+ds/doc/Normaliz.tex-9096-
normaliz-3.8.9+ds/doc/Normaliz.tex:9097:The example program is a simplified version of the program on which the experiments for the paper ``Quantum jumps of normal polytopes'' by W. Bruns, J. Gubeladze and M. Micha\l{}ek, Discrete Comput.\ Geom.\ 56 (2016), no. 1, 181--215, are based. Its goal is to find a maximal normal lattice polytope $P$ in the following sense: there is no normal lattice polytope $Q\supset P$ that has exactly one more lattice point than $P$. `Normal'' means in this context that the Hilbert basis of the cone over $P$ is given by the lattice points of $P$, considered as degree $1$ elements in the cone.
normaliz-3.8.9+ds/doc/Normaliz.tex-9098-