In our earlier study [LR] of prime ideal principles in commutative rings, we have introduced the notion of Oka and Ako families of ideals, along with their “strong analogues”. The logical hierarchy between these ideal families (and the classically well-known monoidal families of ideals) was partly worked out in [LR] over general commutative rings. In the present paper, we amplify this study by ...

Notation 1.1. Let l/k be a finite Galois extension of complete discretely valued fields. Let Ok, Ol, πk, πl, k̄, and l̄ be rings of integers, uniformizers, and residue fields, respectively. For an element a ∈ Ol, we use ā to denote its reduction in l̄. Let G = Gl/k be the Galois group. Use vl(·) to denote the valuation on l so that vl(πl) = 1. We call e = vl(πk) the näıve ramification degree; it i...

The solutions to the Kadomtsev-Petviashvili equation that arise from a fixed complex algebraic curve are parametrized by threefold in weighted projective space, which we name after Boris Dubrovin. Current methods nonlinear algebra applied study parametrizations and defining ideals of Dubrovin threefolds. We highlight dichotomy between transcendental representations exact computations. Our main ...

1 (Dubrovin–Novikov Hamiltonian operator [1]) is compatible with a nondegenerate local Hamiltonian operator of hydrodynamic type K 2 if and only if the operator K 1 is locally the Lie derivative of the operator K 2 along a vector field in the corresponding domain of local coordinates. This result gives, first of all, a convenient general invariant criterion of the compatibility for the Dubrovin...

Classically, Gröbner bases are computed by first prescribing a fixed monomial order. Moss Sweedler suggested an alternative in the mid 1980s and developed a framework to perform such computations by using valuation rings in place of monomial orders. We build on these ideas by providing a class of valuations on K(x, y) that are suitable for this framework. We then perform such computations for i...

Over any field an implicit linear difference equation one can reduce to the usual explicit one, which has infinitely many solutions ~ for each initial value. It is interesting consider over ring, because case of a ring significantly different from one. The previous results on equations rings mostly concern integers and low order equations. In present article high some other classes rings, parti...

We study regular non-semisimple Dubrovin-Frobenius manifolds in dimensions 2,3,4. focus on the case where Jordan canonical form of operator multiplication by Euler vector field has a single block. Our results rely existence special local coordinates introduced [4] for flat F-manifolds with field. In such invariant metric manifold takes which is starting point our construction.

Let K be a field equipped with a valuation. Tropical varieties over K can be defined with a theory of Gröbner bases taking into account the valuation of K. While generalizing the classical theory of Gröbner bases, it is not clear how modern algorithms for computing Gröbner bases can be adapted to the tropical case. Among them, one of the most efficient is the celebrated F5 Algorithm of Faugère....

It is proved that a commutative ring is clean if and only if it is Gelfand with a totally disconnected maximal spectrum. It is shown that each indecomposable module over a commutative ring R satisfies a finite condition if and only if R P is an artinian valuation ring for each maximal prime ideal P. Commutative rings for which each indecomposable module has a local endomorphism ring are studied...

Classically, Gröbner bases are computed by first prescribing a set monomial order. Moss Sweedler suggested an alternative and developed a framework to perform such computations by using valuation rings in place of monomial orders. We build on these ideas by providing a class of valuations on k(x, y) that are suitable for this framework. For these valuations, we compute ν(k[x, y]∗) and use this ...