A Geometric Index Reduction Method for Implicit Systems of Differential Algebraic Equations

This paper deals with the index reduction problem for the class of quasi-regular DAE systems. It is shown that any of these systems can be transformed to a generically equivalent first order DAE system consisting of a single purely algebraic (polynomial) equation plus an under-determined ODE (that is, a semi-explicit DAE system of differentiation index 1) in as many variables as the order of the input system. This can be done by means of a Kronecker-type algorithm with bounded complexity

Differential Chow Form for Projective Differential Variety

In this paper, a generic intersection theorem in projective differential algebraic geometry is presented. Precisely, the intersection of an irreducible projective differential variety of dimension d>0 and order h with a generic projective differential hyperplane is shown to be an irreducible projective differential variety of dimension d-1 and order h. Based on the generic intersection theorem, the Chow form for an irreducible projective differential variety is defined and most of the properties of the differential Chow form in affine differential case are established for its projective differential counterpart. Finally, we apply the differential Chow form to a result of linear dependence over projective varieties given by Kolchin.Comment: 17 page

Multiple Factorizations of Bivariate Linear Partial Differential Operators

We study the case when a bivariate Linear Partial Differential Operator (LPDO) of orders three or four has several different factorizations. We prove that a third-order bivariate LPDO has a first-order left and right factors such that their symbols are co-prime if and only if the operator has a factorization into three factors, the left one of which is exactly the initial left factor and the right one is exactly the initial right factor. We show that the condition that the symbols of the initial left and right factors are co-prime is essential, and that the analogous statement "as it is" is not true for LPDOs of order four. Then we consider completely reducible LPDOs, which are defined as an intersection of principal ideals. Such operators may also be required to have several different factorizations. Considering all possible cases, we ruled out some of them from the consideration due to the first result of the paper. The explicit formulae for the sufficient conditions for the complete reducibility of an LPDO were found also

On globally nilpotent differential equations

In a previous work of the authors, a middle convolution operation on the category of Fuchsian differential systems was introduced. In this note we show that the middle convolution of Fuchsian systems preserves the property of global nilpotence. This leads to a globally nilpotent Fuchsian system of rank two which does not belong to the known classes of globally nilpotent rank two systems. Moreover, we give a globally nilpotent Fuchsian system of rank seven whose differential Galois group is isomorphic to the exceptional simple algebraic group of type $G_2.

Scattering amplitudes in YM and GR as minimal model brackets and their recursive characterization

Attached to both Yang-Mills and General Relativity about Minkowski spacetime are distinguished gauge independent objects known as the on-shell tree scattering amplitudes. We reinterpret and rigorously construct them as $L_\infty$ minimal model brackets. This is based on formulating YM and GR as differential graded Lie algebras. Their minimal model brackets are then given by a sum of trivalent (cubic) Feynman tree graphs. The amplitudes are gauge independent when all internal lines are off-shell, not merely up to $L_\infty$ isomorphism, and we include a homological algebra proof of this fact. Using the homological perturbation lemma, we construct homotopies (propagators) that are optimal in bringing out the factorization of the residues of the amplitudes. Using a variant of Hartogs extension for singular varieties, we give a rigorous account of a recursive characterization of the amplitudes via their residues independent of their original definition in terms of Feynman graphs (this does neither involve so-called BCFW shifts nor conditions at infinity under such shifts). Roughly, the amplitude with $N$ legs is the unique section of a sheaf on a variety of $N$ complex momenta whose residues along a finite list of irreducible codimension one subvarieties (prime divisors) factor into amplitudes with less than $N$ legs. The sheaf is a direct sum of rank one sheaves labeled by helicity signs. To emphasize that amplitudes are robust objects, we give a succinct list of properties that suffice for a dgLa so as to produce the YM and GR amplitudes respectively.Comment: 51 page

Extensions of differential representations of SL(2) and tori

Linear differential algebraic groups (LDAGs) measure differential algebraic dependencies among solutions of linear differential and difference equations with parameters, for which LDAGs are Galois groups. The differential representation theory is a key to developing algorithms computing these groups. In the rational representation theory of algebraic groups, one starts with SL(2) and tori to develop the rest of the theory. In this paper, we give an explicit description of differential representations of tori and differential extensions of irreducible representation of SL(2). In these extensions, the two irreducible representations can be non-isomorphic. This is in contrast to differential representations of tori, which turn out to be direct sums of isotypic representations.Comment: 21 pages; few misprints corrected; Lemma 4.6 adde