Integer-valued polynomials over discrete valuation domains

Abstract

The classical theorem of Lucas states that binomial polynomials, which form a basis for integer-valud polynomials, satisfy a congruence relation, modulo a prime, related to their digits in the base prime representation. In this thesis, we define the Lucas property in the setting of discrete-valued structures and investigate when and where the Lucas property holds. General criteria are derived for bases of integervalued polynomials in this setting to satisfy the Lucas property. Examples of bases including those of Lagrange type and of Carlitz-like polynomials are worked out. In addition, one of the best known properties of binomial polynomials in the classical case is the Pascal triangle equality, which equates the sum of two binomial coefficients to the one in the following line. In the second part of the thesis, we define a general Pascal property and prove a characterization for polynomials which satisfy this Pascal property. Examples of bases of integer-valued polynomials satisfying such a Pascal property, which embrace the classical case, are derived.