Matrices with cyclically monotone rows and Cantor numeration systems

Abstract: We study a class of square matrices with non-negative elements which have cyclically monotone rows in the sense that each row of a matrix from the class consists of a cyclically non-increasing sequence of numbers starting from a maximal element on the diagonal. We prove that if every diagonal element is strictly larger than all other elements in the respective row, then the matrix is regular. This property enables us to solve an open problem that comes from the theory of non-standard numeration systems, also called Cantor numeration systems. The problem concerns a one-to-one relationship between Cantor real bases, which are supposed to be alternate, that is, periodic with a period p, and lists of p sequences of non-negative integers satisfying the so-called Parry condition.

• The paper establishes a one-to-one correspondence between alternate Cantor bases and lists satisfying the Parry condition using cyclically monotone matrices.
• It proves that matrices with strictly cyclically monotone rows have non-negative determinants, ensuring the regularity and injectivity of the power series map.
• Algebraic, combinatorial, and topological methods are integrated to resolve the existence and uniqueness challenges in non-standard Cantor numeration systems.

Authoritative Summary: Matrices with Cyclically Monotone Rows and Cantor Numeration Systems

Introduction and Context

The paper "Matrices with cyclically monotone rows and Cantor numeration systems" (2604.22344) explores a specialized class of square matrices characterized by cyclically monotone rows, building a rigorous bridge between matrix theory and the theory of Cantor numeration systems, with particular emphasis on real and alternate (periodic) bases. The primary motivation is to resolve an open problem regarding the one-to-one correspondence between alternate Cantor real bases of period $p$ and lists of $p$ sequences of non-negative integers satisfying the Parry condition. The analysis intertwines tools from algebra, combinatorics, and topology, establishing new results with strong theoretical implications in numeration system research.

Cyclically Monotone Matrices: Definitions and Main Properties

The paper defines a square matrix $A$ of order $p$ as having cyclically monotone rows if all elements are non-negative and, for every row $j$, the entries $a_{j,j+k}$ (with modular arithmetic) form a cyclically non-increasing sequence beginning at the diagonal element $a_{j,j}$. The stricter condition requires $a_{j,j} > a_{j,j+1}$ for all $j$, establishing strictly cyclically monotone rows. This structure generalizes properties of circulant matrices under monotonicity constraints.

Key invariance properties are established, showing that simultaneous cyclic permutation of rows and columns preserves monotonicity and determinant. Linear combinations and principal submatrices of cyclically monotone matrices inherit this property under non-negativity constraints.

The principal theorem asserts that any matrix with cyclically monotone rows has non-negative determinant, and strictly monotone matrices are regular (i.e., have strictly positive determinant). The proof relies on induction, manipulating rows via algebraic and combinatorial arguments, and leveraging monotonicity to ensure unimodality in submatrices.

Cantor Numeration Systems and the Parry Condition

The Cantor numeration system generalizes base-$\beta$ numeral systems, replacing the fixed base with a sequence $p$0, allowing the representation of numbers in $p$1 via Cantor series:

$p$2

The real Cantor system further relaxes bases to real numbers $p$3 under a growth condition on the product, and alternate bases are those with pure period $p$4. For such systems, the characterization of admissible representations relies on the Parry condition: the $p$5-tuple of sequences $p$6 must satisfy boundedness, non-triviality ($p$7 and $p$8 infinitely often), and for all $p$9, $A$0.

Previously, existence and, in specific cases, uniqueness of alternate bases corresponding to lists of sequences satisfying the Parry condition were established, but the general uniqueness result remained open.

Alternate Power Series and Jacobian Injection

The paper introduces alternate power series functions of the type

$A$1

with $A$2, $A$3, and $A$4 infinitely often. The vector-valued function $A$5 built from $A$6 such series is shown to have a Jacobian matrix whose rows are strictly cyclically monotone when multiplied by a diagonal scaling. Corollary 3.1 asserts that every principal minor of this Jacobian is strictly positive.

Applying the Gale–Nikaido theorem, which states that mappings whose Jacobian has strictly positive principal minors are injective, the paper proves that $A$7 is globally injective on $A$8. This result is central: it enables mapping between sequences and bases in a one-to-one fashion, underpinning uniqueness in the alternate Cantor numeration system problem.

Resolution of Existence and Uniqueness for Alternate Bases

By constructing $A$9 for sequences in a list satisfying the Parry condition and showing surjectivity onto $p$0 (using topological arguments, including Brouwer’s invariance of domain theorem), the paper confirms the existence of a base for any such list.

For uniqueness, the injectivity of $p$1 ensures that two alternate bases yielding the same list of quasi-greedy expansions of unity must coincide. The proof formalizes this via the explicit equation relating the bases to the sequences, relying on the unique solution for $p$2 in $p$3.

Numerical and Structural Results

The strongest result is the proof that strictly cyclically monotone matrices are regular, and more generally, that the determinant is non-negative for cyclically monotone matrices, generalizing properties of monotone circulant matrices. The injectivity result (global univalence) for the vector-valued alternate power series map is bold and contradicts the potential for non-uniqueness in the correspondence between alternate bases and Parry-sequence lists.

Implications and Theoretical Outlook

The results rigorously formalize the connection between complex numeration systems and matrix-theoretic properties, broadening the landscape of allowable bases and sequences in positional numeral systems. Practically, this clarifies the foundational structure for representing numbers using alternate Cantor bases, impacting algorithmic encoding and analytic studies of expansion uniqueness and admissibility.

Theoretically, the methods suggest new approaches for injectivity and univalence in multivariate maps with structured Jacobians and motivate further exploration of non-periodic, non-standard bases. The topological and algebraic tools advocated here could drive broader investigations in number representation and matrix regimes.

Future directions include generalizing the criteria for irrationality, extending Parry-type conditions to infinite lists in non-periodic real Cantor bases, and characterizing deeper algebraic properties of numeration systems through principal minors and global injectivity analyses.

Conclusion

The paper systematically solves the open problem concerning the existence and uniqueness of alternate Cantor real bases corresponding to lists of sequences satisfying the Parry condition, by leveraging properties of matrices with cyclically monotone rows and the topology of alternate power series mappings. The results establish a robust theoretical foundation for Cantor numeration systems, with ramifications for number representation and related algebraic structures.