Related papers: On the Intermediate Value Theorem over a Valued Fi…

The paper proves that all power series over a maximal ordered Cauchy complete non-Archimedean field satisfy the intermediate value theorem on every closed interval. Hensel's Lemma for restricted power series is the main tool of the proof.

We prove an intermediate value theorem of an arithmetical flavor, involving the consecutive averages of sequences with terms in a given finite set A. For every such set we completely characterize the numbers x ("intermediate values") with…

We show that hereditarily indecomposable spaces can be characterized by a special instance of the Intermediate Value Theorem in their rings of continuous functions.

If a real-valued function is continuous on a real interval and it takes on two different values, then it will also take any value in between those two, by the Intermediate Value Theorem. It is not immediately clear what would be a natural…

Let (K, v) be a henselian valued field of arbitrary rank. In this paper, we give an irreducibility criterion for multivariate polynomials over K using valuation theory.

We give a criterion for maps on ultrametric spaces to be surjective and to preserve spherical completeness. We show how Hensel's Lemma and the multi-dimensional Hensel's Lemma follow from our result. We give an easy proof that the latter…

In the present paper a new mean value theorem for polynomials of special form is obtained. The case of sums on vertices of a regular polygon is studied. A criterion for a certain equation to be satisfied is obtained.

We obtain finite field analogues of a series of recent results on various mean value theorems for Weyl sums. Instead of the Vinogradov Mean Value Theorem, our results rest on the classical argument of Mordell, combined with several other…

We discuss the conjecture that every maximal Hardy field has the Intermediate Value Property for differential polynomials, and its equivalence to the statement that all maximal Hardy field are elementarily equivalent to the differential…

This paper proves the approximate intermediate value theorem, constructively and from notably weak hypotheses: from pointwise rather than uniform continuity, without assuming that reals are presented with rational approximants, and without…

The paper gives a unified and simple proof of both theorems and Cousin's theorem.

Let $k$ be a function field of one variable over a finite field with the characteristic not equal to two. In this paper, we consider the prehomogeneous representation of the space of binary quadratic forms over $k$. We have two main…

We consider the problem of uniform interpolation of functions with values in a complex inner product space of finite dimension. This problem can be casted within a modified weighted pluripotential theoretic framework. Indeed, in the…

Proofs of the fundamental theorem of algebra can be divided up into three groups according to the techniques involved: proofs that rely on real or complex analysis, algebraic proofs, and topological proofs. Algebraic proofs make use of the…

We introduce a notion of valued module which is suitable to study valued fields of positive characteristic. Then we built-up a robust theory of henselianity in the language of valued modules and prove Ax-Kochen Ershov type results.

Valuation based systems verifying an idempotent property are studied. A partial order is defined between the valuations giving them a lattice structure. Then, two different strategies are introduced to represent valuations: as infimum of…

We give a first-order definition of key polynomials, we show the links with previous definitions, that it is relevant to study key degrees, and to use a kind of valuations that we call partially multiplicative. We also prove or reprove…

We give an elementary proof of a version of the implicit function theorem over Henselian valued fields $K$. It yields a density property for such fields (introduced in a joint paper with J. Koll{\'a}r), which is indispensable for ensuring…

We present elements of a theory of translation-invariant integration on finite dimensional vector spaces and on GL_n over a valuation field with local field as residue field. We then discuss the case of an arbitrary algebraic group. This…

We know that a continuous function on a closed interval satisfies the Intermediate Value Property. Likewise, the derivative function of a differentiable function on a closed interval satisfies the IVP property which is known as the Darboux…