Algebraic Independence Research Articles

Hitting Sets for Orbits of Circuit Classes and Polynomial Families

The orbit of an n -variate polynomial \(f({\mathbf {x}})\) over a field \(\mathbb {F}\) is the set \(\text {orb}(f) := \lbrace f(A{\mathbf {x}}+{\mathbf {b}}) : A \in \mathrm{GL}(n,\mathbb {F}) \text{ and } {\mathbf {b}}\in \mathbb {F}^n\rbrace\) . The orbit of a polynomial f is a geometrically interesting subset of the set of affine projections of f . Affine projections of polynomials computable by seemingly weak circuit classes can be quite powerful. For example, the polynomial \(\mathsf {IMM}_{3,d}\) —the \((1,1)\) th entry of a product of d generic \(3 \times 3\) matrices—is computable by a constant-width read-once oblivious algebraic branching program (ROABP), yet every polynomial computable by a size- s general arithmetic formula is an affine projection of \(\mathsf {IMM}_{3,\text {poly}(s)}\) as shown by Ben-or and Cleve [ 12 ]. To our knowledge, no efficient hitting set construction was known for \(\text {orb}(\mathsf {IMM}_{3, d})\) before this work. In this article, we initiate the study of explicit hitting sets for the orbits of polynomials computable by several natural and well-studied circuit classes and polynomial families. In particular, we give quasi-polynomial time hitting sets for the orbits of: Low-individual-degree polynomials computable by commutative ROABPs . This implies quasi-polynomial time hitting sets for the orbits of the elementary symmetric polynomials and the orbits of multilinear sparse polynomials . Multilinear polynomials computable by constant-width ROABPs . This implies a quasi-polynomial time hitting set for the orbits of the family \(\lbrace \mathsf {IMM}_{3,d}\rbrace _{d \in \mathbb {N}}\) . Polynomials computable by constant-depth, constant-occur formulas . This implies quasi-polynomial time hitting sets for the orbits of multilinear depth-4 circuits with constant top fan-in , and also polynomial-time hitting sets for the orbits of the power symmetric polynomials and the sum-product polynomials . Polynomials computable by occur-once formulas . We say a polynomial has low individual degree if the degree of every variable in the polynomial is at most \(\text {poly}(\log n)\) , where n is the number of variables. The first two results are obtained by building upon and strengthening the rank concentration by translation technique of Agrawal, Saha, and Saxena [ 6 ]; the second result additionally uses the merge-and-reduce idea from Forbes and Shpilka [ 30 ] and Forbes, Shpilka, and Saptharishi [ 27 ]. The proof of the third result adapts the algebraic independence-based technique of Agrawal, Saha, Saptharishi, and Saxena [ 5 ] and Beecken, Mittmann, and Saxena [ 11 ] to reduce to the case of constructing hitting sets for the orbits of sparse polynomials. A similar reduction using the Shpilka-Volkovich (SV) generator-based argument in Shpilka and Volkovich [ 90 ] yields the fourth result. The SV generator plays an important role in all four results.

On algebraic independence of some continued fractions

In the present paper, we prove the algebraic independence of a finite number of real continued fractions which have partial quotients that increase rapidly. We then use a general Liouville criteria to justiy the algebraic independence of limits of some real series. We note that these results extend some work of Bundschuh and we use a new and simple method.

Algebraic dependences of meromorphic mappings sharing few moving hyperplanes with truncated multiplicity

In this article, we will prove an algebraic dependence theorem for meromorphic mappings into a complex projective space sharing few moving hyperplanes with different truncated multiplicity. Moreover, we also consider the weaker condition: \[\nu _{(f,{a_i}), \le k_i}\le \nu _{(g,{a_i}), \le k_i} \text{ instead of } \nu _{(f,{a_i}), \le k_i}= \nu_{(g,{a_i}), \le k_i}\] for some moving hyperplanes among the given moving hyperplanes. In order to implement this, besides using the technique reported by S. D. Quang in (Two meromorphic mappings having the same inverse images of some moving hyperplanes with truncated multiplicity, Rocky Mountain J. Math., vol. 52, no. 1, pp. 263–273, 2022) we have to separate the 2n + 2 moving hyperplanes from the given p+1 moving hyperplanes. After that, we count multiples of the intersection of the inverse images of the mappings f and g sharing these moving hyperplanes. Our result is an improvement of many previous results in this topic.

On the algebraic difference independence of Γ and ζ

The paper concerns the problem on algebraic difference independence of two functions which come from two different sets G and F, respectively. The first set contains the gamma function and the other set contains Riemann zeta function. Meanwhile, a result is obtained which generalizes a theorem of Li-Tahir-Gao in [22] from the gamma function to the function in G.

Monodromy of double elliptic logarithms

We determine the relative monodromy group of abelian logarithms with respect to periods in the cases of fibered products of elliptic schemes. This gives rise to a result stronger than a theorem due to Y. André and implies in particular the algebraic independence of the logarithm of any non-torsion section and the periods. We then conjecture an analogous result for the general case of an abelian scheme of arbitrary relative dimension. This generalizes a theorem of Corvaja and Zannier which determines the said group in the case of a single elliptic scheme.

A classification of first order differential equations

Let k be a differential field of characteristic zero with an algebraically closed field of constants. In this article, we provide a classification of first order differential equations over k and study the algebraic dependence of solutions of a given first order differential equation. Our results generalize parts of the work of Noordman et al. [24] and complements the work of Freitag et al. [4].

Effective algebraic independence of values of E-functions

Effective algebraic independence of values of E-functions

Algebraic matroids are almost entropic

Algebraic matroids capture properties of the algebraic dependence among elements of extension fields. Almost entropic matroids have the rank functions arbitrarily well approximated by the entropies of subvectors of random vectors. The former class of matroids is included in the latter. A key argument in the proof is the Lang-Weil bound on the number of points in algebraic varieties.

Monodromie unipotente maximale, congruences “à la Lucas” et indépendance algébrique

Let f ( z ) f(z) be in 1 + z Q [ [ z ] ] 1+z\mathbb {Q}[[z]] and S \mathcal {S} be an infinite set of prime numbers such that, for all p ∈ S p\in \mathcal {S} , we can reduce f ( z ) f(z) modulo p p . We let f ( z ) ∣ p f(z)_{\mid p} denote the reduction of f ( z ) f(z) modulo p p . Generally, when f ( z ) f(z) is D-finite, f ∣ p ( z ) f_{\mid p}(z) is algebraic over F p ( z ) \mathbb {F}_p(z) . It turns out that if f ∣ p ( z ) f_{\mid p}(z) is a solution of a polynomial of the form X − A p ( z ) X p l X-A_p(z)X^{p^l} , we can use this type of equations to obtain results of transcendence and algebraic independence over Q ( z ) \mathbb {Q}(z) . In the present paper, we look for conditions on the differential operators annihilating f ( z ) f(z) to guarantee the existence of these particular equations. Suppose that f ( z ) f(z) is solution of a differential operator H ∈ Q ( z ) [ d / d z ] \mathcal {H}\in \mathbb {Q}(z)[d/dz] having a strong Frobenius structure for all p ∈ S p\in \mathcal {S} and we also suppose that f ( z ) f(z) annihilates a Fuchsian differential operator D ∈ Q ( z ) [ d / d z ] \mathcal {D}\in \mathbb {Q}(z)[d/dz] such that zero is a regular singular point of D \mathcal {D} and the exponents of D \mathcal {D} at zero are equal to zero. Our main result states that, for almost every prime p ∈ S p\in \mathcal {S} , f ∣ p ( z ) f_{\mid p}(z) is solution of a polynomial of the form X − A p ( z ) X p l X-A_p(z)X^{p^l} , where A p ( z ) A_p(z) is a rational function with coefficients in F p \mathbb {F}_p of height less than or equal to C p 2 l Cp^{2l} with C C a positive constant that does not depend on p p . We also study the algebraic independence of these power series over Q ( z ) \mathbb {Q}(z) .

Infinite Algebraic Independence of Some Almost Polyadic Numbers

Infinite Algebraic Independence of Some Almost Polyadic Numbers

Algebraic independence of values of quasi-modular forms

In this paper, we discuss the algebraic independence of special values of quasi-modular forms. We show that for three algebraically independent quasi-modular forms with algebraic Fourier coefficients, if one of e2πiz and their values at z in the upper half plane is a non-zero algebraic number, then the other three numbers are algebraically independent over Q. We also provide a condition for determining when quasi-modular forms are algebraically independent. Moreover, based on the theorem proved in the paper, we prove the algebraic independence of values of derivatives of modular forms with algebraic Fourier coefficients.

Algebraic independence of topological Pontryagin classes

Abstract We show that the topological Pontryagin classes are algebraically independent in the rationalised cohomology of B ⁢ Top ⁢ ( d ) {B\mathrm{Top}(d)} for all d ≥ 4 {d\geq 4} .

Algebraic dependences of meromorphic mappings from complete Kähler manifolds into projective spaces sharing few hyperplanes

In this paper, we give some results on the algebraic dependence of meromorphic mappings from a complete Kähler manifold whose universal covering is biholomorphic to a ball in [Formula: see text] into [Formula: see text] sharing few hyperplanes in subgeneral position with truncated multiplicities to level [Formula: see text], where all zeros with multiplicities greater than certain values do not need to be counted. Our results are generalizations and improvements of the previous results in recent times.

Algebraic independence of the partial derivatives of certain functions with arbitrary number of variables

We construct a complex entire function with arbitrary number of variables which has the following property: The infinite set consisting of all the values of all its partial derivatives of any orders at all algebraic points, including zero components, is algebraically independent. In Section 2 of this paper, we develop a technique involving linear isomorphisms and infinite products to replace the algebraic independence of the values of functions in question with that of functions easier to deal with. In Sections 2 and 3, using the technique together with Mahler’s method, we can reduce the algebraic independence of the infinite set mentioned above to the linear independence of certain rational functions modulo the rational function field of many variables. The latter one is solved by the discussions involving a certain valuation and a generic point in Sections 3 and 4.

On the algebraic differential independence of $$ \Gamma $$ and $$ \zeta $$

On the algebraic differential independence of $$ \Gamma $$ and $$ \zeta $$

Peculiarities of thermal calculation of cross-flow plate air recuperators

Application of modern energy-saving technologies is the most important task in development of construction industry. In the conditions of exhausting decrease of energy consumption of public, administrative buildings due to reduction of transmission heat losses the implementation of mechanical ventilation with recuperation of exhaust air heat becomes a rational way to increase energy efficiency. One of the key problems in the study of plate-type air recuperators is to find the temperature efficiency of the device. The value of this characteristic is influenced by such parameters as the area of the branched heat exchange surface, initial temperatures of heat carriers, volume flow rates, flow rate ratio, type of fins, etc. The purpose of the study is to compare coefficients of thermal efficiency of outside air as a function of Reynolds number obtained by engineering calculation, mathematical modelling and manufacturer’s data for the recuperator ISIS Recover HR-A-05-V-G4-E-1-60. The article substantiates the calculation method using the existing algebraic dependence describing heat transfer in the cross-flow scheme of coolant motion on the basis of new results. An empirical dependence is obtained, which adequately describes the results of numerical simulation using the ANSYS Fluent solver and engineering calculation.

Stability of non-degenerate Ricci-type Palatini theories

We study the stability of theories where the gravitational action has arbitrary algebraic dependence on the three first traces of the Riemann tensor: the Ricci tensor, the co-Ricci tensor, and the homothetic curvature tensor. We collectively call them Ricci-type tensors. We allow arbitrary coupling to matter. We consider the case when the connection is unconstrained, and the cases when either torsion or non-metricity is assumed to vanish. We find which combinations of Ricci-type tensors lead to new degrees of freedom around Minkowski and FLRW space, and when there are ghosts. None of the theories with new degrees of freedom are healthy, except the previously known case when torsion is zero and the action depends only on the Ricci tensor. We find that projective invariance is not a sufficient condition for a theory to be ghost-free.

Constructing Faithful Homomorphisms over Fields of Finite Characteristic

We study the question of algebraic rank or transcendence degree preserving homomorphisms over finite fields. This concept was first introduced by Beecken et al. [ 3 ] and exploited by them, and Agrawal et al. [ 2 ] to design algebraic independence–based identity tests using the Jacobian criterion over characteristic zero fields. An analogue of such constructions over finite characteristic fields was unknown due to the failure of the Jacobian criterion over finite characteristic fields. Building on a recent criterion of Pandey et al. [ 15 ], we construct explicit faithful maps for some natural classes of polynomials in the positive characteristic field setting, when a certain parameter called the inseparable degree of the underlying polynomials is bounded (this parameter is always 1 in fields of characteristic zero). This presents the first generalisation of some of the results of Beecken et al. [ 3 ] and Agrawal et al. [ 2 ] in the positive characteristic setting.

Algebraic independence and linear difference equations

We consider pairs of automorphisms $(\phi,\sigma)$ acting on fields of Laurent or Puiseux series: pairs of shift operators $(\phi\colon x\mapsto x+h\_1, \sigma\colon x\mapsto x+h\_2)$, of $q$-difference operators $(\phi\colon x\mapsto q\_1x$, $\sigma\colon x\mapsto q\_2x)$, and of Mahler operators $(\phi\colon x\mapsto x^{p\_1},\ \sigma\colon x\mapsto x^{p\_2})$. Given a solution $f$ to a linear $\phi$-equation and a solution $g$ to an algebraic $\sigma$-equation, both transcendental, we show that $f$ and $g$ are algebraically independent over the field of rational functions, assuming that the corresponding parameters are sufficiently independent. As a consequence, we settle a conjecture about Mahler functions put forward by Loxton and van der Poorten in 1987. We also give an application to the algebraic independence of $q$-hypergeometric functions. Our approach provides a general strategy to study this kind of question and is based on a suitable Galois theory: the $\sigma$-Galois theory of linear $\phi$-equations.

Higher Ramanujan Equations and Periods of Abelian Varieties

We describe higher dimensional generalizations of Ramanujan’s classical differential relations satisfied by the Eisenstein series E 2 E_2 , E 4 E_4 , E 6 E_6 . Such “higher Ramanujan equations” are given geometrically in terms of vector fields living on certain moduli stacks classifying abelian schemes equipped with suitable frames of their first de Rham cohomology. These vector fields are canonically constructed by means of the Gauss-Manin connection and the Kodaira-Spencer isomorphism. Using Mumford’s theory of degenerating families of abelian varieties, we construct remarkable solutions of these differential equations generalizing ( E 2 , E 4 , E 6 ) (E_2,E_4,E_6) , which are also shown to be defined over Z \mathbf {Z} . This geometric framework taking account of integrality issues is mainly motivated by questions in Transcendental Number Theory regarding an extension of Nesterenko’s celebrated theorem on the algebraic independence of values of Eisenstein series. In this direction, we discuss the precise relation between periods of abelian varieties and the values of the above referred solutions of the higher Ramanujan equations, thereby linking the study of such differential equations to Grothendieck’s Period Conjecture. Working in the complex analytic category, we prove “functional” transcendence results, such as the Zariski-density of every leaf of the holomorphic foliation induced by the higher Ramanujan equations.