On Polynomial Functions over Finite Commutative Rings

Let R be an arbitrary finite commutative local ring. In this paper, we obtain a necessary and sufficient condition for a function over R to be a polynomial function. Before this paper, necessary and sufficient conditions for a function to be a polynomial function over some special finite commutative local rings were obtained.     

Cocycle d'Euler etK2

In this paper we construct a canonical 2-cocycle on the groupP SL(2,k) with values in the Witt groupW(k) of the fieldk. This allows us to produce anatural homomorphism :H ₂(SL(2,k); Z)I ²(k), whereI ²(k) is the square of the fundamental ideal. We prove that this homomorphism is in fact a lift of Milnor's symbol.

     

完备化Witt代数中自然出现的一些非有限阶化李代数的表示

讨论了完备化Witt代数的三类非有限阶化子代数的不可约非权模的实现,并利用它们的表示得到一些有趣的组合恒等式.     

Majoration de sommes trigonométriques modulo pn

We give a generic estimation of trigonometric sums defined over closed sub-schemes with semi-stable reduction of the standard affine scheme modulo pn(n 2). We use Greenberg realisation to reduce to trigonometric sums defined over smooths sub-schemes of a finite product of Witt vectors over the finite field of p elements. Using the cohomological interpretation of this sums over a finite field, the sum is directly related to the Fourier–Deligne transformation of the dual pairs of Witt vectors. We deduce the estimation from the properties of the Fourier–Deligne transformation on simple perverse sheaves and pure sheaves.     

Polynomial functions and permutation polynomials over some finite commutative rings

We extend some classical results on polynomial functions . We prove all results in algebraic methods avoiding any combinatorial calculation. As applications of our methods, we obtain some interesting new results on permutation polynomials in several variables over some finite commutative rings.     

Homology of Nilpotent Subalgebras of the Lie Superalgebra K(1,1).3

We calculate the dimensions of the second homology groups with trivial coefficients of nilpotent subalgebras of the Lie superalgebra , which is the natural superanalog of the Witt algebra. The proof is based on direct calculations of the rank of the differential. As an application, we find deformations of the maximal nilpotent subalgebra in .     

Cup product structure for symplectic K2

For a field F with trivial involution we have the Karoubi L-groups ±1^Ln(F). For 0≤n≤2 these groups are intimately related to subgroups of the classical Witt ring of quadratic forms. -1^L ₂(F) also has a presentation by symbols due to Matsumoto. In terms of this data we make explicit calculations for two cup product maps that appear in the L-theory of fields.     

On natural homomorphisms of Witt rings

We prove that the kernel of the ring homomorphism between the Witt ring of any order of a global field and the Witt ring of the field is a nilideal.