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Register a new user$mathcal C^m$ solutions of semialgebraic or definable equations
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We address the question of whether geometric conditions on the given data can be preserved by a solution in (1) the Whitney extension problem, and (2) the Brenner-Fefferman-Hochster-Kollar problem, both for $mathcal C^m$ functions. Our results involve a certain loss of differentiability. Problem (2) concerns the solution of a system of linear equations $A(x)G(x)=F(x)$, where $A$ is a matrix of functions on $mathbb R^n$, and $F$, $G$ are vector-valued functions. Suppose the entries of $A(x)$ are semialgebraic (or, more generally, definable in a suitable o-minimal structure). Then we find $r=r(m)$ such that, if $F(x)$ is definable and the system admits a $mathcal C^r$ solution $G(x)$, then there is a $mathcal C^m$ definable solution. Likewise in problem (1), given a closed definable subset $X$ of $mathbb R^n$, we find $r=r(m)$ such that if $g:Xtomathbb R$ is definable and extends to a $mathcal C^r$ function on $mathbb R^n$, then there is a $mathcal C^m$ definable extension.
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Semialgebraic splines are functions that are piecewise polynomial with respect to a cell decomposition into sets defined by polynomial inequalities. We study bivariate semialgebraic splines, formulating spaces of semialgebraic splines in terms of graded modules. We compute the dimension of the space of splines with large degree in two extreme cases when the cell decomposition has a single interior vertex. First, when the forms defining the edges span a two-dimensional space of forms of degree n---then the curves they define meet in n^2 points in the complex projective plane. In the other extreme, the curves have distinct slopes at the vertex and do not simultaneously vanish at any other point. We also study examples of the Hilbert function and polynomial in cases of a single vertex where the curves do not satisfy either of these extremes.
The Painleve-IV equation has two families of rational solutions generated respectively by the generalized Hermite polynomials and the generalized Okamoto polynomials. We apply the isomonodromy method to represent all of these rational solutions by means of two related Riemann-Hilbert problems, each of which involves two integer-valued parameters related to the two parameters in the Painleve-IV equation. We then use the steepest-descent method to analyze the rational solutions in the limit that at least one of the parameters is large. Our analysis provides rigorous justification for formal asymptotic arguments that suggest that in general solutions of Painleve-IV with large parameters behave either as an algebraic function or an elliptic function. Moreover, the results show that the elliptic approximation holds on the union of a curvilinear rectangle and, in the case of the generalized Okamoto rational solutions, four curvilinear triangles each of which shares an edge with the rectangle; the algebraic approximation is valid in the complementary unbounded domain. We compare the theoretical predictions for the locations of the poles and zeros with numerical plots of the actual poles and zeros obtained from the generating polynomials, and find excellent agreement.
We study the rational solutions of the Abel equation $x=A(t)x^3+B(t)x^2$ where $A,Bin C[t]$. We prove that if $deg(A)$ is even or $deg(B)>(deg(A)-1)/2$ then the equation has at most two rational solutions. For any other case, an upper bound on the number of rational solutions is obtained. Moreover, we prove that if there are more than $(deg(A)+1)/2$ rational solutions then the equation admits a Darboux first integral.
The singularly perturbed Riccati equation is the first-order nonlinear ODE $hbar partial_x f = af^2 + bf + c$ in the complex domain where $hbar$ is a small complex parameter. We prove an existence and uniqueness theorem for exact solutions with prescribed asymptotics as $hbar to 0$ in a halfplane. These exact solutions are constructed using the Borel-Laplace method; i.e., they are Borel summations of the formal divergent $hbar$-power series solutions. As an application, we prove existence and uniqueness of exact WKB solutions for the complex one-dimensional Schrodinger equation with a rational potential.
We study the holographic dual to $c$-extremization for 2d $(0,2)$ superconformal field theories (SCFTs) that have an AdS$_3$ dual realized in Type IIB with varying axio-dilaton, i.e. F-theory. M/F-duality implies that such AdS$_3$ solutions can be mapped to AdS$_2$ solutions in M-theory, which are holographically dual to superconformal quantum mechanics (SCQM), obtained by dimensional reduction of the 2d SCFTs. We analyze the corresponding map between holographic $c$-extremization in F-theory and $mathcal{I}$-extremization in M-theory, where in general the latter receives corrections relative to the F-theory result.
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