A smooth intersection Y of two quadrics in $$\mathbb {P}^{2g+1}$$ has Hodge level 1. We show that such varieties have a multiplicative Chow–Künneth decomposition, the sense Shen–Vial. As consequence, certain tautological subring Chow ring powers injects into cohomology.

Andrew V. Sutherland A key ingredient to improving the efficiency of elliptic curve primality proving (and many other algorithms) is the ability to directly construct an elliptic curve E/Fq with a specified number of rational points, rather than generating curves at random until a suitable curve is found. To do this we need to develop the theory of complex multiplication. Recall from Lecture 7 ...

Given a natural number n, Waring’s problem asks for the minimum natural number sn such that every natural number can be represented as the sum of sn nth powers of integers. In this paper, we will answer this question for the case n = 2. To do this, we will examine the properties of two rings, one of which is a subring of the complex numbers, the other of which is a subring of the quaternions. T...

Proposition 1. Let R be a ring and M an R-module. Then EndR(M), the set of R-linear maps from M to M , is a subring of End(M). Proof. Recall from 3.1.6 that the ring (End(M), +, ◦) is defined by (α + β)(m) = α(m) + β(m), (α ◦ β)(m) = α(β(m)). We must verify the four subring conditions from 3.2.2. (i) The additive identity of End(M) is z : M → M, m → 0M . Note that for any m,n ∈ M z(m + n) = 0M ...

Let A be a finitary algebra over a finite field k, and A-mod the category of finite dimensional left A-modules. Let H(A) be the corresponding Hall algebra, and for a positive integer r let Dr(A) be the subspace of H(A) which has a basis consisting of isomorphism classes of modules in A-mod with at least r+1 indecomposable direct summands. If A is hereditary of type An, then Dr(A) is known to be...

In SCIS 2017, Choi and Kim introduced the new linearly homomorphic ring signature scheme (CK17 scheme) based on the hardness of SIS problem, which overcomes the limitation of Boneh and Freeman’s scheme to implement homomorphic signatures to the real world scenario under multiple signers setting for a message. They replace the original sampling algorithm SamplePre() by Gentry et al. with Wang an...

An ideal K of R is a subset that is both a left ideal and a right ideal of R. For emphasis, we sometimes call it a two-sided ideal but the reader should understand that unless qualified, the word ideal will always refer to a two-sided ideal. The zero ideal (0) and the whole ring R are examples of two-sided ideals in any ring R. A (left)(right) ideal I such that I 6= R is called a proper (left)(...

Sometimes the derivation is denoted by a→ a′. A ring equipped with a derivation is called a differential ring. The notions, differential subring, differential ring extension and homomorphism between differential rings, are clear. If (R, ∂) is a differential ring then the set of constants is by definition {r ∈ R : ∂(r) = 0}, and is easily seen to be a (differential) subring. ∂ means the k-fold i...

On a Lebesgue measure space with measure element do, and total measure finite or infinite, we consider the complex-valued measurable functions f, y, so, ., each determined only a.e., and each belonging to all L,-classes simultaneously, 1<r< a. We donote (i) by F = {f} a ring of functions with complex constants and ordinary multiplication and closed under the involution: if f -= f + if2 e F then...

2. Since A[b] is a subring of B, it is an integral domain. Thus if bz = 0 and b = 0, then z = 0. 3. Any linear transformation on a finite-dimensional vector space is injective iff it is surjective. Thus if b ∈ B and b = 0, there is an element c ∈ A[b] ⊆ B such that bc = 1. Therefore B is a field. 4. Since P is the preimage of Q under the inclusion map of A into B, P is a prime ideal. The map a ...