Related papers: Linear equations on real algebraic surfaces

We give necessary and sufficient existence criteria, and methods for finding, continuous solutions of linear equations whose coefficients are polynomials.

We show that a linear functional equation with polynomial coefficients need not admit an arc-analytic solution even if it admits a continuous semialgebraic one. We also show that such an equation need not admit a Nash regulous solution even…

The Lie algebra of planar vector fields with coefficients from the field of rational functions over an algebraically closed field of characteristic zero is considered. We find all finite-dimensional Lie algebras that can be realized as…

A general formalism to solve nonlinear differential equations is given. Solutions are found and reduced to those of second order nonlinear differential equations in one variable. The approach is uniformized in the geometry and solves…

We consider an ordinary nonlinear differential equation with generalized coefficients as an equation in differentials in algebra of new generalized functions. Then the solution of such equation will be a new generalized function. In the…

We show that a proof in multiplicative linear logic can be represented as a decorated surface, such that two proofs are logically equivalent just when their surfaces are geometrically equivalent. This is an extended abstract for…

This note studies, and partially solves, 3 elementary questions about continuous rational functions on real (and p-adic) algebraic varieties: Can one restrict such a function to a subvariety? Can one extend such a function from a…

It is classical that univariate algebraic functions satisfy linear differential equations with polynomial coefficients. Linear recurrences follow for the coefficients of their power series expansions. We show that the linear differential…

Modelling real world systems frequently requires the solution of systems of nonlinear equations. A number of approaches have been suggested and developed for this computational problem. However, it is also possible to attempt solutions…

Linear differential equations of arbitrary order with polynomial coefficients are considered. Specifically, necessary and sufficient conditions for the existence of polynomial solutions of a given degree are obtained for these equations. An…

We prove, for a wide class of semilinear elliptic differential and pseudodifferential equations in $\R^d$, that the solutions which are sufficiently regular and have a certain decay at infinity extend to holomorphic functions in sectors of…

Segre proved that a smooth cubic surface over Q is unirational iff it has a rational point. We prove that the result also holds for cubic hypersurfaces over any field, including finite fields.

The aim of this article is to show that systems of linear partial differential equations on filtered manifolds, which are of weighted finite type, can be canonically rewritten as first order systems of a certain type. This leads immediately…

A continuous solution of an algebraic equation with holomorphic almost periodic coefficients is also almost periodic.

For a linear difference equation with the coefficients being computable sequences, we establish algorithmic undecidability of the problem of determining the dimension of the solution space including the case when some additional prior…

We show by finding an explicit parametrization that a 4th degree surface which arises as a necessary condition for the existence of a perfect cuboid is a rational surface, i.e. birationally equivalent over $\mathbb Q$ to a plane.

For any positive integer $r$, we construct a smooth complex projective rational surface which has at least $r$ real forms not isomorphic over $\mathbb{R}$.

Starting from the results of Charles Fefferman and Janos Koll\`ar in Continuous Solutions of Linear Equations [1], we adopt a new approach based on Fefferman's techniques of Glaeser refinement to show a more general result than the one…

A rational perfect cuboid is a rectangular parallelepiped whose edges and face diagonals are given by rational numbers and whose space diagonal is equal to unity. Its existence is equivalent to the existence of a perfect cuboid with all…

We conjecture recurrence relations satisfied by the degrees of some linearizable lattice equations. This helps to prove linear growth of these equations. We then use these recurrences to search for lattice equations that have linear growth…