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Register a new userAn Analytic Approach to The Cancellation Problem for Affine Spaces over $mathbb{C}$
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Authors
Susumu Oda
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The Cancellation Problem for Affine Spaces is settled affirmatively, that is, it is proved that : Let $ k $ be an algebraically closed field of characteristic zero and let $n, m in mathbb{N}$. If $R[Y_1,..., Y_m] cong_k k[X_1,..., X_{n+m}]$ as $k$-algebras, where $Y_1,..., Y_m, X_1,..., X_{n+m}$ are indeterminates, then $R cong_k k[X_1,..., X_n]$.
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We have proved the following Problem:{it Let $R$ be a $mathbb{C}$-affine domain, let $T$ be an element in $R setminus mathbb{C}$ and let $i : mathbb{C}[T] hookrightarrow R$ be the inclusion. Assume that $R/TR cong_{mathbb{C}} mathbb{C}^{[n-1]}$ and that $R_T cong_{mathbb{C}[T]} mathbb{C}[T]_T^{[n-1]}$. Then $R cong_{mathbb{C}} mathbb{C}^{[n]}$.} This result leads to the negative solution of the candidate counter-example of V.Arno den Lessen : Conjecture E : {it Let $A:=mathbb{C}[t,u,x,y,z]$ denote a polynomial ring, and let $f(u):=u^3-3u, g(u):=u^4-4u^2$ and $h(u):=u^5-10u$ be the polynomials in $mathbb{C}[u]$. Let $D:= f(u)partial_x + g(u)partial_y + h(u)partial_z + tpartial_u$ (which is easily seen to be a locally nilpotent derivation on $A$). Then $A^D otcong_{mathbb{C}} mathbb{C}^{[4]}$.} Consequently our result in this short paper guarantees that the conjectures : the Cancellation Problem for affine spaces, the Linearization Problem, the Embedding Problem and the affine $mathbb{A}^n$-Fibration Problem are still open.
We prove estimates for the level of distribution of the Mobius function, von Mangoldt function, and divisor functions in squarefree progressions in the ring of polynomials over a finite field. Each level of distribution converges to $1$ as $q$ goes to $infty$, and the power savings converges to square-root cancellation as $q$ goes to $infty$. These results in fact apply to a more general class of functions, the factorization functions, that includes these three. The divisor estimates have applications to the moments of $L$-functions, and the von Mangoldt estimate to one-level densities.
Let K be a finite field and let X* be an affine algebraic toric set parameterized by monomials. We give an algebraic method, using Groebner bases, to compute the length and the dimension of C_X*(d), the parameterized affine code of degree d on the set X*. If Y is the projective closure of X*, it is shown that C_X^*(d) has the same basic parameters that C_Y(d), the parameterized projective code on the set Y. If X* is an affine torus, we compute the basic parameters of C_X*(d). We show how to compute the vanishing ideals of X* and Y.
We provide a number of new conjectures and questions concerning the syzygies of $mathbb{P}^1times mathbb{P}^1$. The conjectures are based on computing the graded Betti tables and related data for large number of different embeddings of $mathbb{P}^1times mathbb{P}^1$. These computations utilize linear algebra over finite fields and high-performance computing.
In this note we study and obtain factorization theorems for colorings of matrices and Grassmannians over $mathbb{R}$ and ${mathbb{C}}$, which can be considered metr
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