New Maths Tool Brings Infinite Functions under Control for Better Modelling

Scientists have long sought to understand log canonical thresholds, crucial invariants in complex geometry, and this research addresses a global version of this concept for plurisubharmonic functions exhibiting logarithmic growth in complex spaces. Carles Bivià-Ausina of Universitat Politècnica de València and Alexander Rashkovskii of University of Stavanger, along with colleagues, present explicit formulas for the toric case and establish a novel polynomial approximation of these functions, offering improved control over singularities and behaviour at infinity, a global extension of Demailly’s approximation. This work significantly advances the field by providing tools for analysing plurisubharmonic functions and has implications for the study of tame polynomial maps, potentially unlocking new insights into their properties.

This work introduces a method for describing the range of functions where the exponential of a negative function, e−u, belongs to the Lp space for all p greater than zero.

Explicit formulas for this threshold, denoted as c∞(u), have been derived specifically for toric functions, offering a detailed analysis of their behaviour. The study centres on a novel polynomial approximation of plurisubharmonic functions, controlling both singularities and behaviour at infinity, representing a global extension of Demailly’s approximation theorem.
By examining Bergman functions within corresponding weighted Hilbert spaces, the researchers developed a technique to approximate these functions with increased precision and control. This advancement allows for a more accurate representation of complex functions as they extend towards infinity. This research establishes a notion of log canonical threshold at infinity, c∞(u), defined as the greatest lower bound for which e−cu is square-integrable away from a compact set.

Its reciprocal, λ∞(u), represents the integrability index at infinity. While in one dimension, λ∞(u) corresponds to the logarithmic type of u, the situation becomes more complex in higher dimensions, requiring careful consideration of the limits of integrability. The team demonstrated that for toric functions, c∞(u) can be computed using formulas analogous to those of Kiselman and Howald, leveraging the indicator diagram of u.

Furthermore, the study investigates multipliers at infinity, polynomials that affect the integrability of e−u, revealing that they worsen integrability compared to their local counterparts. The collection of these multipliers, denoted P∞(u), forms a vector space with specific properties related to the indicator diagram of u.

A Hilbert space is constructed using polynomials satisfying certain integrability conditions, leading to the development of a Bergman kernel function and a convergence result for approximating u, controlling both local singularities and behaviour at infinity. This convergence is achieved through the analysis of Bergman kernel functions associated with Hilbert spaces formed by polynomials, offering a powerful tool for studying plurisubharmonic functions.

Bergman function analysis and polynomial approximation of plurisubharmonic functions are central to complex analysis

Researchers investigated a global definition of log canonicality for plurisubharmonic functions exhibiting logarithmic growth, aiming to characterise the range of such functions. The study commenced with detailed analysis of Bergman functions associated with corresponding weighted Hilbert spaces, enabling the construction of a novel polynomial approximation for plurisubharmonic functions of logarithmic growth.

This approximation provides precise control over singularities and asymptotic behaviour, representing a global extension of Demailly’s approximation technique. Central to the methodology was the explicit computation of formulas within the toric case, facilitating a deeper understanding of the behaviour of these functions in specific geometric settings.

The work then established a new method for approximating plurisubharmonic functions, carefully managing both their singular points and their behaviour as the variable approaches infinity. This innovative approach builds upon and generalises existing techniques for polynomial approximation, offering improved control over the approximation’s properties.

Furthermore, the research explored applications to tame polynomial maps, demonstrating the utility of the developed theoretical framework in a related area of complex analysis. The precise control over singularities achieved through the polynomial approximation proved crucial in analysing the behaviour of these maps.

This methodology allowed for a rigorous examination of the log canonical threshold, a key concept in the study of singularities and their impact on algebraic geometry. The findings contribute to a more refined understanding of plurisubharmonic functions and their role in various mathematical contexts.

Canonical thresholds and logarithmic types of plurisubharmonic functions on complex spaces are important tools in complex analysis

Logarithmic canonical thresholds for plurisubharmonic functions of logarithmic growth are considered, aiming to describe the range of all such functions. Explicit formulas were obtained in the toric case, and a new polynomial approximation of plurisubharmonic functions with logarithmic growth was established, providing control over singularities and behavior at infinity.

This represents a global version of Demailly’s approximation and has applications to tame polynomial maps. For functions belonging to the class L∗(Cn), the value of c∞(u+v) is less than or equal to the minimum of c∞(u) and c∞(v) for any u, v in L∗(Cn). Additionally, c∞(tu) equals t−1c∞(u) for all t greater than zero, and c∞(C) equals positive infinity for any constant C.

Proposition 5.1 establishes that c∞(u+v) ≤ c∞(max{u, v}) ≤ min{c∞(u), c∞(v)} for any u, v ∈ L∗(Cn). Corollary 5.2 demonstrates that for any u ∈ L∗(Cn), 0 is less than n σu ≤ c∞(u) ≤ n L∞(u) ≤ positive infinity, where σu represents the logarithmic type and L∞(u) is the Łojasiewicz exponent at infinity.

Specifically, for any map P ∈ P∗, n deg P ≤ c∞(P) ≤ n L∞(P). The lower bounds c∞(u) ≥ c∞(u) ≥ n σu are valid for any u ∈ L(Cn), and e−u+ is in L2(Cn) if σu ≤ n. Skoda’s inequalities show that the integrability index λζ(u) at any point ζ ∈ Cn is comparable to the Lelong number νζ(u), while at infinity, the logarithmic type σu plays the role of the Lelong number.

Corollary 5.2 establishes that λ∞(u) ≤ σu/n, though a uniform lower bound for λ∞(u) is not guaranteed even within the class L∗(Cn). For vN(z) = log max{|z1|, |zN 2|, |z1zN 2|, 1}, λ∞(vN) = 1 and σvN = N + 1. For Ψ ∈ L∗(Cn), the integral of e−2cΨ+ βn converges if and only if c sup P k tk ψ(t) : t ∈ H+, where H+ is defined by X k tk 0.

Consequently, c∞(Ψ) = 1 inf{ψ(t) : t ∈ H1}, meaning that the point c−11 lies in the interior of the indicator diagram ΓΨ+ of Ψ+. Proposition 6.1 confirms this, and Lemma 6.2 proves that Z Rn e−hK(t) dt 0 for all t ≠ 0, where hK is the supporting function of a bounded convex body K ⊂ Rn. Given a = (a1, . . . , an) ∈ Rn +, c∞(Sa) = X k ak, where Sa(z) = max k log |zk| ak.

Polynomial approximation of plurisubharmonic functions with logarithmic growth and controlled asymptotic behaviour is a challenging problem

Researchers have established a new polynomial approximation for plurisubharmonic functions exhibiting logarithmic growth, with precise control over singularities and behaviour at infinity. This advancement builds upon a global version of Demailly’s approximation, extending its capabilities to functions of this specific type.

Explicit formulas have been derived for the toric case, facilitating a deeper understanding of these functions within that geometric context. The work centres on a global notion of log canonicality for plurisubharmonic functions with logarithmic growth, determining the range of such functions. By utilising Bergman functions associated with weighted Hilbert spaces, the study demonstrates a method for approximating these functions with polynomials, ensuring controlled behaviour both locally and as the variable approaches infinity.

Applications to tame polynomial maps are also considered, suggesting potential uses in related areas of complex analysis. The authors acknowledge limitations regarding the applicability of their results to indicators not belonging to a specific class, noting that the established formulas may not hold universally.

Furthermore, the convergence of certain infimums to specific sets is not always guaranteed without additional conditions, such as the indicator diagram being a lower set. Future research could focus on extending these findings to a broader range of indicators and exploring the properties of indicator diagrams in greater detail, potentially refining the criteria for convergence and broadening the scope of the approximation method.