Dimensional Affine Space Research Articles

Left-symmetric algebras and homogeneous improper affine spheres

The nonzero level sets in $n$-dimensional flat affine space of a translationally homogeneous function are improper affine spheres if and only if the Hessian determinant of the function is equal to a nonzero constant multiple of the $n$th power of the function. The exponentials of the characteristic polynomials of certain left-symmetric algebras yield examples of such functions whose level sets are analogues of the generalized Cayley hypersurface of Eastwood-Ezhov. There are found purely algebraic conditions sufficient for the characteristic polynomial of the left-symmetric algebra to have the desired properties. Precisely, it suffices that the algebra has triangularizable left multiplication operators and the trace of the right multiplication is a Koszul form for which right multiplication by the dual idempotent is projection along its kernel, which equals the derived Lie subalgebra of the left-symmetric algebra.

On rational fixed points of finite group actions on the affine space

Consider a finite l-group acting on the affine space of dimension n over a field k, whose characteristic differs from l. We prove the existence of a fixed point, rational over k, in the following cases: --- The field k is p-special for some prime p different from its characteristic. --- The field k is perfect and fertile, and n = 3.

A Differential Analog of the Noether Normalization Lemma

In this paper, we prove the following differential analog of the Noether normalization lemma: for every $d$-dimensional differential algebraic variety over differentially closed field of zero characteristic there exists a surjective map onto the $d$-dimensional affine space. Equivalently, for every integral differential algebra $A$ over differential field of zero characteristic there exist differentially independent $b_1, \ldots, b_d$ such that $A$ is differentially algebraic over subalgebra $B$ differentially generated by $b_1, \ldots, b_d$, and whenever $\mathfrak{p} \subset B$ is a prime differential ideal, there exists a prime differential ideal $\mathfrak{q} \subset A$ such that $\mathfrak{p} = B \cap \mathfrak{q}$. We also prove the analogous theorem for differential algebraic varieties over the ring of formal power series over an algebraically closed differential field and present some applications to differential equations.

An effective study of polynomial maps

In this paper, using an effective algorithm, we obtain an equivalent statement to the Jacobian Conjecture. For a polynomial map [Formula: see text] on an affine space of dimension [Formula: see text] over a field of characteristic [Formula: see text], we define recursively a finite sequence of polynomial maps. We give an equivalent condition to the invertibility of [Formula: see text] as well as a formula for [Formula: see text] in terms of this finite sequence of polynomial maps. Some examples illustrate the effective aspects of our approach.

Some remarks on representations of Yang-Mills algebras

In this article, we present some new properties of representations of Yang-Mills algebras. We first show that any free Lie algebra with m generators is a quotient of the Yang-Mills algebra 𝔶𝔪(n) on n generators, for n ≥ 2m. We derive from this that any semisimple Lie algebra and even any affine Kac-Moody algebra is a quotient of 𝔶𝔪(n) for n ≥ 4. Combining this with previous results on representations of Yang-Mills algebras given in [Herscovich and Solotar, Ann. Math. 173(2), 1043–1080 (2011)], one may obtain solutions to the Yang-Mills equations by differential operators acting on sections of twisted vector bundles on the affine space of dimension n ≥ 4 associated to representations of any semisimple Lie algebra. We also show that this quotient property does not hold for n = 3, since any morphism of Lie algebras from 𝔶𝔪(3) to 𝔰𝔩(2, k) has in fact solvable image.

Small complete caps from singular cubics, II

Small complete arcs and caps in Galois spaces over finite fields $$\mathbb {F}_q$$ F q with characteristic greater than three are constructed from singular cubic curves. For $$m$$ m a divisor of $$q+1$$ q + 1 or $$q-1$$ q - 1 , complete plane arcs of size approximately $$q/m$$ q / m are obtained, provided that $$(m,6)=1$$ ( m , 6 ) = 1 and $$m<\frac{1}{4}q^{1/4}$$ m < 1 4 q 1 / 4 . If in addition $$m=m_1m_2$$ m = m 1 m 2 with $$(m_1,m_2)=1$$ ( m 1 , m 2 ) = 1 , then complete caps in affine spaces of dimension $$N\equiv 0 \pmod 4$$ N ? 0 ( mod 4 ) with roughly $$\frac{m_1+m_2}{m}q^{N/2}$$ m 1 + m 2 m q N / 2 points are described. These results substantially widen the spectrum of $$q$$ q s for which complete arcs in $$AG(2,q)$$ A G ( 2 , q ) of size approximately $$q^{3/4}$$ q 3 / 4 can be constructed. Complete caps in $$AG(N,q)$$ A G ( N , q ) with roughly $$q^{(4N-1)/8}$$ q ( 4 N - 1 ) / 8 points are also provided. For infinitely many $$q$$ q s, these caps are the smallest known complete caps in $$AG(N,q)$$ A G ( N , q ) , $$N \equiv 0 \pmod 4$$ N ? 0 ( mod 4 ) .

Fundamental domains for properly discontinuous affine groups

We construct a fundamental region for the action on the $2d+1$-dimensional affine space of some free, discrete, properly discontinuous groups of affine transformations preserving a quadratic form of signature $(d+1, d)$, where $d$ is any odd positive integer.

On Addition Formulae for Sigma Functions of Telescopic Curves

A telescopic curve is a certain algebraic curve defined by $m-1$ equations in the affine space of dimension $m$, which can be a hyperelliptic curve and an $(n,s)$ curve as a special case. We extend the addition formulae for sigma functions of $(n,s)$ curves to those of telescopic curves. The expression of the prime form in terms of the derivative of the sigma function is also given.

On the structure of the directions not determined by a large affine point set

Given a point set $U$ in an $n$-dimensional affine space of size $q^{n-1}-\varepsilon$, we obtain information on the structure of the set of directions that are not determined by $U$, and we describe an application in the theory of partial ovoids of certain partial geometries.

Bicovering arcs and small complete caps from elliptic curves

Bicovering arcs in Galois affine planes of odd order are a powerful tool for the construction of complete caps in spaces of arbitrarily higher dimensions. The aim of this paper is to investigate whether the arcs contained in elliptic cubic curves are bicovering. As a result, bicovering k-arcs in AG(2,q) of size k≤q/3 are obtained, provided that q−1 has a prime divisor m with 7<m<(1/8)q 1/4. Such arcs produce complete caps of size kq (N−2)/2 in affine spaces of dimension N≡0(mod4). When q=p h with p prime and h≤8, these caps are the smallest known complete caps in AG(N,q), N≡0(mod4).

Effective theories of connections and curvature: Abelian case

We introduce a notion of measuring scales for quantum abelian gauge systems. At each measuring scale a finite dimensional affine space stores information about the evaluation of the curvature on a discrete family of surfaces. Affine maps from the spaces assigned to finer scales to those assigned to coarser scales play the role of coarse graining maps. This structure induces a continuum limit space which contains information regarding curvature evaluation on all piecewise linear surfaces with boundary. The evaluation of holonomies along loops is also encoded in the spaces introduced here; thus, our framework is closely related to loop quantization and it allows us to discuss effective theories in a sensible way. We develop basic elements of measure theory on the introduced spaces which are essential for the applicability of the framework to the construction of quantum abelian gauge theories.

Murphy’s law for Hilbert function strata in the Hilbert scheme of points

An open question is whether the Hilbert scheme of points of a high dimensional affine space satisfies Murphy's Law, as formulated by Vakil. In this short note, we instead consider the loci in the Hilbert scheme parametrizing punctual schemes with a given Hilbert function, and we show that these loci satisfy Murphy's Law. We also prove a related result for equivariant deformations of curve singularities with $\mathbb G_m$-action.

RUSSELL’S HYPERSURFACE FROM A GEOMETRIC POINT OF VIEW

The famous Russell hypersurface is a smooth complex affine threefold which is diffeomorphic to a euclidean space but not algebraically isomorphic to the three dimensional affine space. This fact was first established by Makar-Limanov, using algebraic minded techniques. In this article, we give an elementary argument which adds a greater insight to the geometry behind the original proof and which also may be applicable in other situations.

On the Yao-Yao partition theorem

The Yao-Yao partition theorem states that given a probability measure on an affine space of dimension n having a density which is continuous and bounded away from 0, it is possible to partition the space into 2^n regions of equal measure in such a way that every affine hyperplane avoids at least one of the regions. We give a constructive proof of this result and extend it to slightly more general measures.

On Grassmannizable Group 3-Webs

The authors prove that the Lie group G generating a Grassmannizable group 3-web GGW is the group of parameters of the group of similarity transformations of an (r−1)-dimensional affine space \(\mathbb{A}^{r-1}\) . The transitive action of the group G on itself is an r-parameter subgroup B(r) of the group A(r 2+r) of affine transformations z I =a J I x J +b I ,I,J=1,…,r, which is the direct product of the one-dimensional group of homotheties z 1=kx 1 and r−1 one-dimensional groups of affine transformations \(z^{i}=kx^{i}+b^{i},\;i=2,\ldots ,r,\) where all r groups have the same homothety coefficient k. Conversely, the Lie group B(r) described above generates a Grassmannizable group 3-web GGW. The Lie group G is solvable but not nilpotent.

Configuration spaces of n lines in affine (n + k)-space

In this paper, we study the space of configurations of n lines in the affine space of dimension n + k. We give a topological description of these spaces in terms of some fiber spaces and describe more specifically some interesting cases.

The motivic zeta function and its smallest poles

Let f be a regular function on a nonsingular complex algebraic variety of dimension d. We prove a formula for the motivic zeta function of f in terms of an embedded resolution. This formula is over the Grothendieck ring itself, and specializes to the formula of Denef and Loeser over a certain localization. We also show that the space of n-jets satisfying f = 0 can be partitioned into locally closed subsets which are isomorphic to a cartesian product of some variety with an affine space of dimension ⌜ d n / 2 ⌝ . Finally, we look at the consequences for the poles of the motivic zeta function.

The Jacobian Conjecture is Stably Equivalent to the Dixmier Conjecture

The Jacobian conjecture in dimension $n$ asserts that any polynomial endomorphism of $n$-dimensional affine space over a field of zero characteristic, with the Jacobian equal 1, is invertible. The Dixmier conjecture in rank $n$ asserts that any endomorphism of the $n$-th Weyl algebra (the algebra of polynomial differential operators in $n$ variables) is invertible. We prove that the Jacobian conjecture in dimension $2n$ implies the Dixmier conjecture in rank $n$. Together with a well-known implication in the opposite direction, it shows that the stable Jacobian and Dixmier conjectures are equivalent. The main tool of the proof is the reduction to finite characteristic. After the paper was finished we have learned that the main result was already published by Y.Tsuchimoto in Osaka Journal of Mathematics Volume 42, Number 2 (June 2005). His proof is different.

Gauss-Manin connections for arrangements, IV. Nonresonant eigenvalues

An arrangement is a finite set of hyperplanes in a finite dimensional complex affine space. A complex rank one local system on the arrangement complement is determined by a set of complex weights for the hyperplanes. We study the Gauss-Manin connection for the moduli space of arrangements of fixed combinatorial type in the cohomology of the complement with coefficients in the local system determined by the weights. For nonresonant weights, we solve the eigenvalue problem for the endomorphisms arising in the 1 -form associated to the Gauss-Manin connection.

Determinants of the Hypergeometric Period Matrices of an Arrangement and Its Dual

We fix three natural numbers $k, n, N$, such that $n+k+1=N$, and introduce the notion of two dual arrangements of hyperplanes. One of the arrangements is an arrangement of $N$ hyperplanes in a $k$-dimensional affine space, the other is an arrangement of $N$ hyperplanes in an $n$-dimensional affine space. We assign weights $\al_1, ..., \al_N$ to the hyperplanes of the arrangements and for each of the arrangements consider the associated period matrices. The first is a matrix of $k$-dimensional hypergeometric integrals and the second is a matrix of $n$-dimensional hypergeometric integrals. The size of each matrix is equal to the number of bounded domains of the corresponding arrangement. We show that the dual arrangements have the same number of bounded domains and the product of the determinants of the period matrices is equal to an alternating product of certain values of Euler's gamma function multiplied by a product of exponentials of the weights.