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Gabber's original proof of his purity theorem

It's different, but it also uses Weil II. See Purity for intersection cohomology after Deligne-Gabber for my translation of the original.
• 32.1k
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When does a perverse sheaf occur in the decomposition theorem?

In general it is a difficult problem. For example, the core of Ngô's proof of the fundamental lemma is his support theorem which implies that in the context of the Hitchin fibration, all simple ...
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What is the elementary proof of Weil's polynomial theorem of decomposition?

If $P(x,y),Q(x,y)$ are relatively prime, then so are the one-variable polynomials $p(x)=P(x,1),q(x)=Q(x,1)$ (since we can homogenize any common factor of $p,q$ to a common factor of $P,Q$). It follows ...
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What can be said about a projective morphisms that admit decomposition theorem like smooth morphisms?

I don't know how to characterize such morphisms, which I think is your first question. However, this can certainly happen, even if $f$ is not smooth. (By the way, your comment about absence of local ...
• 32.1k
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What is this matrix decomposition called and does it exist always?

No, you can't always find two such matrices. Let $n \gt 2$. Choose a $2$-dimensional subspace $V$ of $\mathbb{Q}^n$ that does not intersect the positive orthant except at $\vec 0$. Let $M$ be a ...
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Equality or inequality for determinant of $A_{n \times m} D_{m \times m} A^T_{m \times n}$

Yes, there is, using exterior product as a functor: Let $e_i$ be the basis of $\mathbb R^n$. Then \begin{align} \det(A.D.A^\top)e_1\wedge\dots\wedge e_n &= \Lambda^n(A.D.A^\top)(e_1\wedge\dots\...
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Singular Value decomposition of huge dimensional matrix

This problem of updating the singular value decomposition of a matrix upon repeatedly appending a row or a column to the matrix is discussed in A stable and fast algorithm for updating the singular ...
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• 3,865
1 vote

Decomposition theorem over more general base schemes

If $X$ is any excellent scheme with equicodimensional irreducible components, there is a perverse t-structure on $D^b_c(X,\mathbf{Q}_\ell)$ with all expected properties ($\ell$ any prime invertible on ...
• 12.7k
1 vote
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Generate a two-variable polynomial from its "roots

There a number of very well researched techniques that can be used to solve your problem in a practical sense. Most of these have come from computer graphics and computer vision where taking a set of ...
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Find a minimum set of paths that cover all pairs of dependent vertices

An answer via hypergraph theory There exists an obvious reduction to the minimum edge cover problem in hypergraph theory (also known as the set cover problem). I will describe this reduction. A ...
• 5,833
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A Krull-Schmidt Theorem for Lie groups?

I see that you are restricting to (connected) compact Lie groups. This means that $G$ is a product of a semi-simple group and a central torus. The semi-simple piece decomposes uniquely (up to ...
1 vote
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Nonlinear system of equations whereas most of the equations are linear. How to minimise operation?

Let us write your matrix equation as the following block format, $$\begin{bmatrix}A & B\\ C & D(x,y) \end{bmatrix}\begin{bmatrix}x\\ y \end{bmatrix}=\begin{bmatrix}b\\ c \end{bmatrix},$$ in ...
1 vote

Examples for Decomposition Theorem

A nice example of the decomposition theorem for a proper morphism of smooth varieties is the blowup of the points $p_i$ lying at the intersection of two plane curves; that is, let \$f, g \in \mathbb{C}[...
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