By James Matteson
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Additional resources for A collection of diophantine problems with solutions
A tensor category over K is a K-linear category C, with a bi-functor ⊗ : C × C → C and an object 1 ∈ Obj(C) for which there are functorial isomorphisms aX,Y,Z : X ⊗ (Y ⊗ Z) → (X ⊗ Y ) ⊗ Z cX,Y : X ⊗ Y → Y ⊗ X lX : X ⊗ 1 → X and rX : 1 ⊗ X → X. These satisfy consistency conditions expressed in the form of a triangle, a pentagon, and a hexagon diagram, see [Andr´e (2004)] for details. One Motives and periods 33 also generally assumes that commutativity holds, which means that cY,X = c−1 X,Y , although this can be replaced by a signed version.
Under a suitable choice of an equivalence relation on the set of algebraic cycles, one can obtain in this way an abelian category of pure motives and in fact a Tannakian category, with a corresponding Galois group of symmetries, the motivic Galois group. This part of the theory of motives is at this point on solid ground, although problems like explicitly identifying the motivic Galois groups of various significant subcategories of the category of pure motives, or identifying when a correspondence is realized by an algebraic cycle, remain extremely challenging.
For nonnegative integers. The function Γ(t) defined in this way extends to a meromorphic function with poles at all the non-positive integers. As we are going to see in more detail below, a typical way of dealing with divergences in Feynman integrals is to first identify them with poles of some meromorphic function and typically a product of Gamma functions. The first useful operation on Feynman integrals is the introduction of Schwinger parameters. These are based on the very simple identity 1 = q ∞ e−sq ds.
A collection of diophantine problems with solutions by James Matteson