Forms In Several Variables Research Articles

Algebraic methods to study the dimension of supersmooth spline spaces

Multivariate piecewise polynomial functions (or splines) on polyhedral complexes have been extensively studied over the past decades and find applications in diverse areas of applied mathematics including numerical analysis, approximation theory, and computer aided geometric design. In this paper we address various challenges arising in the study of splines with enhanced mixed (super-)smoothness conditions at the vertices and across interior faces of the partition. Such supersmoothness can be imposed but can also appear unexpectedly on certain splines depending on the geometry of the underlying polyhedral partition. Using algebraic tools, a generalization of the Billera–Schenck–Stillman complex that includes the effect of additional smoothness constraints leads to a construction which requires the analysis of ideals generated by products of powers of linear forms in several variables. Specializing to the case of planar triangulations, a combinatorial lower bound on the dimension of splines with supersmoothness at the vertices is presented, and we also show that this lower bound gives the exact dimension in high degree. The methods are further illustrated with several examples.

On the representation of integers by decomposable forms in several variables

On the representation of integers by decomposable forms in several variables

The Poincaré series of the algebra of rational functions which are regular outside hyperplanes

Let Δ be a finite set of nonzero linear forms in several variables with coefficients in a field K of characteristic zero. Consider the K -algebra R( Δ) of rational functions on V which are regular outside ⋃ α∈ Δ ker α. Then the ring R( Δ) is naturally doubly filtered by the degrees of denominators and of numerators. In this paper we give an explicit combinatorial formula for the Poincaré series in two variables of the associated bigraded vector space R(Δ) . This generalizes the main theorem of [H. Terao, Algebras generated by reciprocals of linear forms, J. Algebra 250 (2002) 549–558].

Algebras Generated by Reciprocals of Linear Forms

Let Δ be a finite set of nonzero linear forms in several variables with coefficients in a field K of characteristic zero. Consider the K-algebra C(Δ) of rational functions generated by {1/α∣α∈Δ}. Then the ring ∂(V) of differential operators with constant coefficients naturally acts on C(Δ). We study the graded ∂(V)-module structure of C(Δ). We especially find standard systems of minimal generators and a combinatorial formula for the Poincaré series of C(Δ). Our proofs are based on a theorem by Brion–Vergne [4] and results by Orlik–Terao [9].

THE TOPOLOGICAL q-EXPANSION PRINCIPLE

Imitating the classical q-expansion principle we use the elliptic character map to develop the relation between elements in elliptic cohomology and their q-series in K-theory. We show that, under certain exactness conditions, the integrality of elliptic objects is completely controlled by their characters. As an application, we obtain an interpretation of the cooperations in elliptic cohomology as was conjectured by Clarke and Johnson [11]. It enables us to give a description of the elliptic-based Adams–Novikov spectral sequence in terms of cyclic cohomology of modular forms in several variables, and to set up a higher e-invariant with values in Katz’s [23] ring of divided congruences. We show how the topological q-expansion principle can be used to equip elliptic cohomology with orientations which obey various Riemann–Roch formulas.

Small zeros of additive forms in several variables

Letf(X) be an additive form defined by $$f(X) = f(x_1 ,x_2 , \cdots ,x_3 ) = \sigma _1 a_1 x_{_1 }^k + \sigma _2 a_2 x_2^k + \cdots \sigma _3 a_3 x_{_3 }^k ,$$ whereai≠0 is integer,i=1,2…,s. In 1979, Schmidt proved that if ∈>0 then there is a large constantC(k,∈) such that fors>C(k,∈) the equationf(X)=0 has a nontrivial, integer solution in σ1, σ2, …, σ3,x1,x2, …,x3 satisfying $$\sigma _i = \pm and \left| {x_1 } \right| \leqslant A^e , i = 1,2, \cdots ,s where A = \mathop {max}\limits_{1 \leqslant i \leqslant s} \left| {a_1 } \right|.$$ Schmidt did not estimate this constantC(k,∈) since it would be extremely large. In this paper, we prove the following result

Representations of loop groups, Dirac operators on loop space, and modular forms

FOR M a compact Spin manifold of dimension d= 2k, the unicersal elliptic genus p(M), in the sense of D. V. Chudnovsky, G. V. Chudnovsky, P. Landweber, S. Ochanine, and R. E. Stong, is a modular form of weight k for a level 2 congruence subgroup of SL(2, Z) [SJ, [17], [19], [26], and [28]. E. Witten showed that p(M) could be interpreted as the index of a sort of twisted Dirac operator on the loop manifold LM [26] and [27]. He actually considered several sorts of twistings of this Dirac operator; all of them give rise to modular forms for level 2 congruence subgroups. Similar results were obtained by Alvarez, Killingback, Mangano, and Windey [l] and by Schellekens and Warner [23]. In this article we consider, in a purely formal way, the index of the Dirac operator on LM, with coefficients in the vector bundle Reassociated to a “positive energy” representation of the universal central extension e Spin(d) of the loop group LSpin(d). We do not claim to understand anything about this operator, except that its index may be formally written down. The main result (Theorem 2.2) is that this index is a modular form of weight k for a congruence subgroup of SL(2, Z), which depends on the level of the representation of zSpin(d). The proof uses the results of V. KaE and D. Peterson [ll] and [12] on the characters of such representations, in particular the fact they are Jacobi modular forms in several variables. We present a conjecture relating the representations of tSpin(d) with elliptic cohomology, the theory of Landweber, Ravenel, and Stong [17], [19]. More precisely, we extend elliptic cohomology to higher levels (i.e. for each integer N, we construct a homology theory Ellz such that Ellt(pt) is the ring of modular forms for the principal congruence subgroup F(N), whose Fourier expansions at every cusp have coefficients in Z [ e2ni’N]). The point of this construction is that Ellz(pt) is a faithfully flat module over Ell,(pt); so we do not claim that Ellt(pt) is a really new homology theory, from the topological veiwpoint. From the differential-geometric viewpoint, we show that the vector bundle E, mentioned above, is equivariant under the Virasoro algebra, which acts on LM through its quotient vecr(S’). We think it might be useful to view the index of the Dirac operator on LM as a virtual representation of the Virasoro algebra, rather than merely the group of rotations of S’.

Application of symbolic manipulation to Hecke transformations of modular forms in two variables

The Hecke transformation of modular forms in several variables generates nonsymmetric modular forms out of symmetric forms. This is useful since symmetric forms arise out of Eisenstein series and are easy to construct, while nonsymmetric forms are much harder to construct. A symbolic manipulation system is required because of the magnitude of the Fourier expansions. This process is carried out for Hilbert modular functions over Q ( 2 ) \mathbb {Q}(\sqrt 2 ) .

Application of Symbolic Manipulation to Hecke Transformations of Modular Forms in Two Variables

The Hecke transformation of modular forms in several variables generates nonsymmetric modular forms out of symmetric forms. This is useful since symmetric forms arise out of Eisenstein series and are easy to construct, while nonsymmetric forms are much harder to construct. A symbolic manipulation system is required because of the magnitude of the Fourier expansions. This process is carried out for Hilbert modular functions over $\mathbb {Q}(\sqrt 2 )$.

Arithmetic Properties of Fourier-Jacobi Expansions of Automorphic Forms in Several Variables

In this paper we establish some basic arithmetic results concerning automorphic forms in several variables. Our results are, in part, analogues of the classical facts for elliptic modular forms, and results of Shimura on Hilbert and Siegel modular forms, [9] and [10]. In essence, this involves relating rationality properties of Fourier-Jacobi coefficients to rationality properties in algebraic geometry. Our automorphic forms are C-valued holomorphic functions on certain Cartesian powers Hnd of Siegel upper half-spaces Hn. We consider appropriate arithmetic groups r acting on Hnd such that r\H,d is noncompact. In particular, we reduce our study to that of congruence subgroups r of reductive linear algebraic groups G such that the derived group G' is almost simple over Q. Further, let K be a maximal compact subgroup of G'(R)+, the connected component of the identity in the real

A lattice-point problem for norm forms in several variables

Following Mahler's work “Zur Approximation algebraischer Zahlen,” 1 1 Part I, in Math. Ann. 107 (1933), 691–730; Part II, Math. Ann. 108 (1933), 37–55; Part III, Acta Math. 62 (1934), 91–166. an asymptotic formula for the number of lattice points in | Φ( x 0,…, x h )| ≦ k, where Φ is a norm form of a certain type in h + 1 variables, is obtained as k → ∞. The work is however also based on the author's paper “Approximation of Algebraic Numbers.” 2 2 Göttingen Nachrichten 1966, 45–52.