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Markov numbers are integers that appear in the solution triples of the Diophantine equation, $x^2+y^2+z^2=3xyz$, called the Markov equation. A classical topic in number theory, these numbers are related to many areas of mathematics such as combinatorics, hyperbolic geometry, approximation theory and cluster algebras. There is a natural map from the rational numbers between zero and one to the Markov numbers. In this paper, we prove two conjectures seen in Martin Aigner's book, Markov's theorem and 100 years of the uniqueness conjecture, that determine an ordering on subsets of the Markov numbers based on their corresponding rational. The proof relies on a relationship between Markov numbers and continuant polynomials which originates in Frobenius' 1913 paper., 16 pages

Let $X \subseteq \mathbb{P}^r$ be a non-degenerate smooth projective variety of dimension $n$, codimension $e$, and degree $d$ defined over an algebraically closed field of characteristic zero. In this paper, we first show that $\text{reg} (\mathcal{O}_X) \leq d-e$, and classify the extremal and the next to extremal cases. Our result reduces the Eisenbud-Goto regularity conjecture for the smooth case to the problem finding a Castelnuovo-type bound for normality. It is worth noting that McCullough-Peeva recently constructed counterexamples to the regularity conjecture by showing that $\text{reg} (\mathcal{O}_X)$ is not even bounded above by any polynomial function of $d$ when $X$ is not smooth. For a normality bound in the smooth case, we establish that $\text{reg}(X) \leq n(d-2)+1$, which improves previous results obtained by Mumford, Bertram-Ein-Lazarsfeld, and Noma. Finally, by generalizing Mumford's method on double point divisors, we prove that $\text{reg}(X) \leq d-1+m$, where $m$ is an invariant arising from double point divisors associated to outer general projections. Using double point divisors associated to inner projection, we also obtain a slightly better bound for $\text{reg}(X)$ under suitable assumptions., Comment: 23 pages. This paper has been largely rewritten after McCullough-Peeva's counterexamples to the Eisenbud-Goto regularity conjecture, which appeared in J. Amer. Math. Soc. in 2018. We also added new results on the regularity of smooth projective varieties of arbitrary dimension

We study the hit problem, set up by F. Peterson, of finding a minimal set of generators for the polynomial algebra $P_k := \mathbb F_2[x_1,x_2,...,x_k]$ as a module over the mod-2 Steenrod algebra, $\mathcal{A}$. In this paper, we study a minimal set of generators for $\mathcal A$-module $P_k$ in some so-call generic degrees and apply these results to explicitly determine the hit problem for $k=4$., Comment: 68 pages, Quy Nhon University preprint, Viet Nam, 2011. A shorter version of this paper was published in Advances in Mathematics

We present an algebraic framework which simultaneously generalizes the notion of linear subspaces, matroids, valuated matroids, oriented matroids, and regular matroids. To do this, we first introduce algebraic objects called tracts which generalize both hyperfields in the sense of Krasner and partial fields in the sense of Semple and Whittle. We then define matroids over tracts; in fact, there are (at least) two natural notions of matroid in this general context, which we call weak and strong matroids. We give "cryptomorphic" axiom systems for such matroids in terms of circuits, Grassmann-Pl\"ucker functions, and dual pairs, and establish some basic duality results. We then explore sufficient criteria for the notions of weak and strong matroids to coincide. For example, if $F$ is a particularly nice kind of tract called a doubly distributive partial hyperfield, we show that the notions of weak and strong $F$-matroids coincide. We also give examples of tracts $F$ and weak $F$-matroids which are not strong. Our theory of matroids over tracts is closely related to, but more general than, "matroids over fuzzy rings" in the sense of Dress and Dress-Wenzel., Comment: 35 pages. v2: Final version to appear in Advances in Mathematics. Numerous minor updates and revisions. v1: This paper generalizes and subsumes the results of arXiv:1601.01204. Our treatment now includes partial fields, for example, in addition to hyperfields

In this paper, we give the sharp upper bound for the number of vertices with positive curvature in a planar graph with nonnegative combinatorial curvature. Based on this, we show that the automorphism group of a planar---possibly infinite---graph with nonnegative combinatorial curvature and positive total curvature is a finite group, and give an upper bound estimate for the order of the group., 32 pages, 19 figures

Let $d\geq 2$ be an integer and let $2d/(d-1) < q \leq \infty$. In this paper we investigate the sharp form of the mixed norm Fourier extension inequality \begin{equation*} \big\|\widehat{f\sigma}\big\|_{L^q_{{\rm rad}}L^2_{{\rm ang}}(\mathbb{R}^d)} \leq {\bf C}_{d,q}\, \|f\|_{L^2(\mathbb{S}^{d-1},{\rm d}\sigma)}, \end{equation*} established by L. Vega in 1988. Letting $\mathcal{A}_d \subset (2d/(d-1), \infty]$ be the set of exponents for which the constant functions on $\mathbb{S}^{d-1}$ are the unique extremizers of this inequality, we show that: (i) $\mathcal{A}_d$ contains the even integers and $\infty$; (ii) $\mathcal{A}_d$ is an open set in the extended topology; (iii) $\mathcal{A}_d$ contains a neighborhood of infinity $(q_0(d), \infty]$ with $q_0(d) \leq \left(\tfrac{1}{2} + o(1)\right) d\log d$. In low dimensions we show that $q_0(2) \leq 6.76\,;\,q_0(3) \leq 5.45 \,;\, q_0(4) \leq 5.53 \,;\, q_0(5) \leq 6.07$. In particular, this breaks for the first time the even exponent barrier in sharp Fourier restriction theory. The crux of the matter in our approach is to establish a hierarchy between certain weighted norms of Bessel functions, a nontrivial question of independent interest within the theory of special functions., Comment: 21 pages

Let $S$ be a polynomial ring over a field and $I\subseteq S$ a homogeneous ideal containing a regular sequence of forms of degrees $d_1, \ldots, d_c$. In this paper we prove the Lex-plus-powers Conjecture when the field has characteristic 0 for all regular sequences such that $d_i \geq \sum_{j=1}^{i-1} (d_j-1)+1$ for each $i$; that is, we show that the Betti table of $I$ is bounded above by the Betti table of the lex-plus-powers ideal of $I$. As an application, when the characteristic is 0, we obtain bounds for the Betti numbers of any homogeneous ideal containing a regular sequence of known degrees, which are sharper than the previously known ones from the Bigatti-Hulett-Pardue Theorem., To appear in Advances in Mathematics

Fa\`a di Bruno's formula gives an expression for the derivatives of the composition of two real-valued functions. In this paper we prove a multivariate and synthesized version of Fa\`a di Bruno's formula in higher dimensions, providing a combinatorial expression for the derivatives of chain compositions $F^{(1)} \circ \ldots \circ F^{(m)}$ of functions $F^{(l)} : \mathbb{R}^N \to \mathbb{R}^N$ in terms of sums over labelled trees. We give several applications of this formula, including a new involution formula for the inversion of multivariate power series. We use this framework to outline a combinatorial approach to studying the invertibility of polynomial mappings, giving a purely combinatorial restatement of the Jacobian conjecture. Our methods extend naturally to the non-commutative case, where we prove a free version of Fa\`a di Bruno's formula for multivariate power series in free indeterminates, and use this formula as a tool for obtaining a new inversion formula for free power series., Comment: 38 pages, 11 figures

Let $G$ be a finite connected graph, and let $\rho$ be the spectral radius of its universal cover. For example, if $G$ is $k$-regular then $\rho=2\sqrt{k-1}$. We show that for every $r$, there is an $r$-covering (a.k.a. an $r$-lift) of $G$ where all the new eigenvalues are bounded from above by $\rho$. It follows that a bipartite Ramanujan graph has a Ramanujan $r$-covering for every $r$. This generalizes the $r=2$ case due to Marcus, Spielman and Srivastava (2013). Every $r$-covering of $G$ corresponds to a labeling of the edges of $G$ by elements of the symmetric group $S_{r}$. We generalize this notion to labeling the edges by elements of various groups and present a broader scenario where Ramanujan coverings are guaranteed to exist. In particular, this shows the existence of richer families of bipartite Ramanujan graphs than was known before. Inspired by Marcus-Spielman-Srivastava, a crucial component of our proof is the existence of interlacing families of polynomials for complex reflection groups. The core argument of this component is taken from a recent paper of them (2015). Another important ingredient of our proof is a new generalization of the matching polynomial of a graph. We define the $r$-th matching polynomial of $G$ to be the average matching polynomial of all $r$-coverings of $G$. We show this polynomial shares many properties with the original matching polynomial. For example, it is real rooted with all its roots inside $\left[-\rho,\rho\right]$., Comment: 38 pages, 4 figures, journal version (minor changes from previous arXiv version). Shortened version appeared in STOC 2016

Let $G$ be connected reductive algebraic group defined over an algebraically closed field of characteristic $p > 0$ and suppose that $p$ is a good prime for the root system of $G$, the derived subgroup of $G$ is simply connected and the Lie algebra $\mathfrak{g} = \operatorname{Lie}(G)$ admits a non-degenerate Ad$(G)$-invariant symmetric bilinear form. Given a linear function $\chi$ on $\mathfrak{g}$ we denote by $U_\chi(\mathfrak{g})$ the reduced enveloping algebra of $\mathfrak{g}$ associated with $\chi$. By the Kac-Weisfeiler conjecture (now a theorem), any irreducible $U_\chi(\mathfrak{g})$-module has dimension divisible by $p^{d(\chi)}$ where $2d(\chi)$ is the dimension of the coadjoint $G$-orbit containing $\chi$. In this paper we give a positive answer to the natural question raised in the 1990s by Kac, Humphreys and the first-named author and show that any algebra $U_\chi(\mathfrak{g})$ admits a module of dimension $p^{d(\chi)}$., Comment: The present version is accepted for publication

Let $(R,m)\to (S,n)$ be a flat local extension of local rings. Lech conjectured in 1960 that there should be a general inequality $e(R)\leq e(S)$ on the Hilbert-Samuel multiplicities. This conjecture is known when the base ring $R$ has dimension less than or equal to two, and remains open in higher dimensions. In this paper, we prove Lech's conjecture in dimension three when $R$ has equal characteristic. In higher dimension, our method yields substantial partial estimate: $e(R)\leq (d!/2^d)\cdot e(S)$ where $d=\dim R\geq 4$, in equal characteristic., Final version

Let $K$ be a self-similar set satisfying the open set condition. Following Kaimanovich's elegant idea, it has been proved that on the symbolic space $X$ of $K$ a natural augmented tree structure ${\mathfrak E}$ exists; it is hyperbolic, and the hyperbolic boundary $\partial_HX$ with the Gromov metric is H\"older equivalent to $K$. In this paper we consider certain reversible random walks with return ratio $0< \lambda, Comment: 33 pages with 2 figures

In this paper we explore the generalized Nash problem for arcs on a germ of smooth surface: given two prime divisors above its special point, to determine whether the arc space of one of them is included in the arc space of the other one. We prove that this problem is combinatorial and we explore its relation with several notions of adjacency of plane curve singularities., Comment: Final version

Let $W\in \mathbb{C}[x_1,\cdots,x_N]$ be an invertible polynomial with an isolated singularity at origin, and let $G\subset {{\sf SL}}_N\cap (\mathbb{C}^*)^N$ be a finite diagonal and special linear symmetry group of $W$. In this paper, we use the category ${{\sf MF}}_G(W)$ of $G$-equivariant matrix factorizations and its associated VSHS to construct a $G$-equivariant version of Saito's theory of primitive forms. We prove there exists a canonical categorical primitive form of ${{\sf MF}}_G(W)$ characterized by $G_W^{{\sf max}}$-equivariance. Conjecturally, this $G$-equivariant Saito theory is equivalent to the genus zero part of the FJRW theory under LG/LG mirror symmetry. In the marginal deformation direction, we verify this for the FJRW theory of $\big(\frac{1}{5}(x_1^5+\cdots+x_5^5),\mathbb{Z}/5\mathbb{Z}\big)$ with its mirror dual B-model Landau-Ginzburg orbifold $\big(\frac{1}{5}(x_1^5+\cdots+x_5^5), (\mathbb{Z}/5\mathbb{Z})^4\big)$. In the case of the Quintic family $\mathcal{W}=\frac{1}{5}(x_1^5+\cdots+x_5^5)-\psi x_1x_2x_3x_4x_5$, we also prove a comparison result of B-model VSHS's conjectured by Ganatra-Perutz-Sheridan., Comment: 30 pages, comments welcome! Minor modification in V2

The main result of this paper is the following. Given countably many multivariate polynomials with rational coefficients and maximum degree d, we construct a compact set E ⊂ R n of Hausdorff dimension n / d which does not contain finite point configurations corresponding to the zero sets of the given polynomials. Given a set E ⊂ R n , we study the angles determined by three-point subsets of E. The main result implies the existence of a compact set in R n of Hausdorff dimension n / 2 which does not contain the angle π / 2 . (This is known to be sharp if n is even.) We show that there is a compact set of Hausdorff dimension n / 8 which does not contain an angle in any given countable set. We also construct a compact set E ⊂ R n of Hausdorff dimension n / 6 for which the set of angles determined by E is Lebesgue null. In the other direction, we present a result that every set of sufficiently large dimension contains an angle e close to any given angle. The main result can also be applied to distance sets. As a corollary we obtain a compact set E ⊂ R n ( n ≥ 2 ) of Hausdorff dimension n / 2 which does not contain rational distances nor collinear points, for which the distance set is Lebesgue null, moreover, every distance and direction is realised only at most once by E.

In 1958, Sierpinski asked whether there exists a linear order X that is isomorphic to its lexicographically ordered cube but is not isomorphic to its square. The main result of this paper is that the answer is negative. More generally, if X is isomorphic to any one of its finite powers X n , n > 1 , it is isomorphic to all of them.

Bennett, Carbery and Tao established nearly optimal $L^1$ trilinear restriction estimates in $\mathbb{R}^{n+1}$ under transversality assumptions only. In this paper we show that the curvature improves the range of exponents, by establishing $L^p$ estimates, for any $p > \frac{2(n+4)}{3(n+2)}$ in the case of double-conic surfaces. The exponent $\frac{2(n+4)}{3(n+2)}$ is shown to be the universal threshold for the trilinear estimate., arXiv admin note: text overlap with arXiv:1601.01000

Let $\mathrm{R}$ be a real closed field. The problem of obtaining tight bounds on the Betti numbers of semi-algebraic subsets of $\mathrm{R}^k$ in terms of the number and degrees of the defining polynomials has been an important problem in real algebraic geometry with the first results due to Ole{\u��}nik and Petrovski{\u��}, Thom and Milnor. These bounds are all exponential in the number of variables $k$. Motivated by several applications in real algebraic geometry, as well as in theoretical computer science, where such bounds have found applications, we consider in this paper the problem of bounding the equivariant Betti numbers of symmetric algebraic and semi-algebraic subsets of $\mathrm{R}^k$. We obtain several asymptotically tight upper bounds. In particular, we prove that if $S\subset \mathrm{R}^k$ is a semi-algebraic subset defined by a finite set of $s$ symmetric polynomials of degree at most $d$, then the sum of the $\mathfrak{S}_k$-equivariant Betti numbers of $S$ with coefficients in $\mathbb{Q}$ is bounded by $(skd)^{O(d)}$. Unlike the classical bounds on the ordinary Betti numbers of real algebraic varieties and semi-algebraic sets, the above bound is polynomial in $k$ when the degrees of the defining polynomials are bounded by a constant. As an application we improve the best known bound on the ordinary Betti numbers of the projection of a compact algebraic set improving for any fixed degree the best previously known bound for this problem due to Gabrielov, Vorobjov and Zell., Minor changes. Final version to appear in Advances in Mathematics

Let V(0) be the mod 2 Moore spectrum and let C be the supersingular elliptic curve over F_4 defined by the Weierstrass equation y^2+y=x^3. Let F_C be its formal group law and E_C be the spectrum classifying the deformations of F_C. The group of automorphisms of F_C, which we denote by S_C, acts on E_C. Further, S_C admits a surjective homomorphism to the 2-adic integers whose kernel we denote by S_C^1. The cohomology of S_C^1 with coefficients in (E_C)_*V(0) is the E_2-term of a spectral sequence converging to the homotopy groups of the homotopy fix points of E_C smash V(0) with respect to S_C^1, a spectrum closely related to L_{K(2)}V(0). In this paper, we use the algebraic duality resolution spectral sequence to compute an associated graded for H^*(S_C^1;(E_C)_*V(0)). These computations rely heavily on the geometry of elliptic curves made available to us at chromatic level 2., Comment: Largely rewritten and re-organized. Added an appendix on the cohomology of G_{24} based on notes of Hans-Werner Henn. Appeared in Adv. Math

Several important conjectures in Fractal Geometry can be summarised as follows: If the dimension of a self-similar measure in R does not equal its expected value, then the underlying iterated function system contains an exact overlap. In recent years significant progress has been made towards these conjectures. Hochman proved that if the Hausdorff dimension of a self-similar measure in R does not equal its expected value, then there are cylinders which are super-exponentially close at all small scales. Several years later, Shmerkin proved an analogous statement for the L q dimension of self-similar measures in R . With these statements in mind, it is natural to wonder whether there exist iterated function systems that do not contain exact overlaps, yet there are cylinders which are super-exponentially close at all small scales. In this paper we show that such iterated function systems do exist. In fact we prove much more. We prove that for any sequence ( ϵ n ) n = 1 ∞ of positive real numbers, there exists an iterated function system { ϕ i } i ∈ I that does not contain exact overlaps and min ⁡ { | ϕ a ( 0 ) − ϕ b ( 0 ) | : a , b ∈ I n , a ≠ b , r a = r b } ≤ ϵ n for all n ∈ N .

Based on results by Brugall\'e and Mikhalkin, Fomin and Mikhalkin give formulas for computing classical Severi degrees $N^{d, \delta}$ using long-edge graphs. In 2012, Block, Colley and Kennedy considered the logarithmic version of a special function associated to long-edge graphs appeared in Fomin-Mikhalkin's formula, and conjectured it to be linear. They have since proved their conjecture. At the same time, motivated by their conjecture, we consider a special multivariate function associated to long-edge graphs that generalizes their function. The main result of this paper is that the multivariate function we define is always linear. A special case of our result gives an independent proof of Block-Colley-Kennedy's conjecture. The first application of our linearity result is that by applying it to classical Severi degrees, we recover quadraticity of $Q^{d, \delta}$ and a bound $\delta$ for the threshold of polynomiality of $N^{d, \delta}.$ Next, in joint work with Osserman, we apply the linearity result to a special family of toric surfaces and obtain universal polynomial results having connections to the G\"ottsche-Yau-Zaslow formula. As a result, we provide combinatorial formulas for the two unidentified power series $B_1(q)$ and $B_2(q)$ appearing in the G\"ottsche-Yau-Zaslow formula. The proof of our linearity result is completely combinatorial. We define $\tau$-graphs which generalize long-edge graphs, and a closely related family of combinatorial objects we call $(\tau, n)$-words. By introducing height functions and a concept of irreducibility, we describe ways to decompose certain families of $(\tau, n)$-words into irreducible words, which leads to the desired results., Comment: 38 pages, 1 figure, 1 table. Major revision: generalized main results in previous version. The old results only applies to classical Severi degrees. The current version also applies to Severi degrees coming from special families of toric surfaces

Since the early 2000s physicists have developed an ingenious but non-rigorous formalism called the cavity method to put forward precise conjectures on phase transitions in random problems (Mezard et al., 2002 [37] ). The cavity method predicts that the satisfiability threshold in the random k-SAT problem is r k - SAT = 2 k ln ⁡ 2 − 1 2 ( 1 + ln ⁡ 2 ) + e k , with lim k → ∞ ⁡ e k = 0 (Mertens et al., 2006 [35] ). This paper contains a proof of the conjecture.

In a recent paper [Adv. Math. 305:165--196, 2017], Komornik et al.~proved a long-conjectured formula for the Hausdorff dimension of the set $\mathcal{U}_q$ of numbers having a unique expansion in the (non-integer) base $q$, and showed that this Hausdorff dimension is continuous in $q$. Unfortunately, their proof contained a gap which appears difficult to fix. This article gives a completely different proof of these results, using a more direct combinatorial approach., Comment: 18 pages

In this paper, we answer a question of Dwyer, Greenlees, and Iyengar by proving a local ring $R$ is a complete intersection if and only if every complex of $R$-modules with finitely generated homology is proxy small. Moreover, we establish that a commutative noetherian ring $R$ is locally a complete intersection if and only if every complex of $R$-modules with finitely generated homology is virtually small., Comment: 14 pages

Building on earlier papers of several authors, we establish that there exists a universal constant $c > 0$ such that the minimal base size $b(G)$ of a primitive permutation group $G$ of degree $n$ satisfies $\log |G| / \log n \leq b(G) < 45 (\log |G| / \log n) + c$. This finishes the proof of Pyber's base size conjecture. An ingredient of the proof is that for the distinguishing number $d(G)$ (in the sense of Albertson and Collins) of a transitive permutation group $G$ of degree $n > 1$ we have the estimates $\sqrt[n]{|G|} < d(G) \leq 48 \sqrt[n]{|G|}$., 27 pages, referee comments included

Let $(\hat{X}, T^{1,0} \hat{X})$ be a compact orientable CR embeddable three dimensional strongly pseudoconvex CR manifold, where $T^{1,0} \hat{X}$ is a CR structure on $\hat{X}$. Fix a point $p \in \hat{X}$ and take a global contact form $\hat{\theta}$ so that $\hat{\theta}$ is asymptotically flat near $p$. Then $(\hat{X}, T^{1,0} \hat{X}, \hat{\theta})$ is a pseudohermitian $3$-manifold. Let $G_p \in C^{\infty} (\hat{X} \setminus \{p\})$, $G_p > 0$, with $G_p(x) \sim \vartheta(x,p)^{-2}$ near $p$, where $\vartheta(x,y)$ denotes the natural pseudohermitian distance on $\hat{X}$. Consider the new pseudohermitian $3$-manifold with a blow-up of contact form $(\hat{X} \setminus \{p\}, T^{1,0} \hat{X}, G^2_p \hat{\theta})$ and let $\Box_{b}$ denote the corresponding Kohn Laplacian on $\hat{X} \setminus \{p\}$. In this paper, we prove that the weighted Kohn Laplacian $G^2_p \Box_b$ has closed range in $L^2$ with respect to the weighted volume form $G^2_p \hat{\theta} \wedge d\hat{\theta}$, and that the associated partial inverse and the Szeg\"{o} projection enjoy some regularity properties near $p$. As an application, we prove the existence of some special functions in the kernel of $\Box_{b}$ that grow at a specific rate at $p$. The existence of such functions provides an important ingredient for the proof of a positive mass theorem in 3-dimensional CR geometry by Cheng-Malchiodi-Yang., Comment: 72 pages

For each dimension $d$, $d$-dimensional integral simplices with exactly one interior integral point have bounded volume. This was first shown by Hensley. Explicit volume bounds were determined by Hensley, Lagarias and Ziegler, Pikhurko, and Averkov. In this paper we determine the exact upper volume bound for such simplices and characterize the volume-maximizing simplices. We also determine the sharp upper bound on the coefficient of asymmetry of an integral polytope with a single interior integral point. This result confirms a conjecture of Hensley from 1983. Moreover, for an integral simplex with precisely one interior integral point, we give bounds on the volumes of its faces, the barycentric coordinates of the interior integral point and its number of integral points. Furthermore, we prove a bound on the lattice diameter of integral polytopes with a fixed number of interior integral points. The presented results have applications in toric geometry and in integer optimization., Advances in Mathematics

In this paper we prove the Bannai-Ito conjecture, namely that there are only finitely many distance-regular graphs of fixed valency greater than two., 51 pages

This paper examines dimension of the graph of the famous Weierstrass non-differentiable function \[ W_{\lambda, b} (x) = \sum_{n=0}^{\infty}\lambda^n\cos(2\pi b^n x) \] for an integer $b \ge 2$ and $1/b < \lambda < 1$. We prove that for every $b$ there exists (explicitly given) $\lambda_b \in (1/b, 1)$ such that the Hausdorff dimension of the graph of $W_{\lambda, b}$ is equal to $D = 2+\frac{\log\lambda}{\log b}$ for every $\lambda\in(\lambda_b,1)$. We also show that the dimension is equal to $D$ for almost every $\lambda$ on some larger interval. This partially solves a well-known thirty-year-old conjecture. Furthermore, we prove that the Hausdorff dimension of the graph of the function \[ f (x) = \sum_{n=0}^{\infty}\lambda^n\phi(b^n x) \] for an integer $b \ge 2$ and $1/b < \lambda < 1$ is equal to $D$ for a typical $\mathbb Z$-periodic $C^3$ function $\phi$., Comment: Final authors' version with a correction of an inexact statement in the introduction to the published version, concerning the box dimension of the graphs of functions of the form (1.1) and (1.2)

This paper is devoted to the approximation of differentiable semialgebraic functions by Nash functions. Approximation by Nash functions is known for semialgebraic functions defined on an affine Nash manifold M, and here we extend it to functions defined on Nash subsets X of M whose singularities are monomial. To that end we discuss first "finiteness" and "weak normality" for such sets X. Namely, we prove that (i) X is the union of finitely many open subsets, each Nash diffeomorphic to a finite union of coordinate linear varieties of an affine space and (ii) every function on X which is Nash on every irreducible component of X extends to a Nash function on M. Then we can obtain approximation for semialgebraic functions and even for certain semialgebraic maps on Nash sets with monomial singularities. As a nice consequence we show that m-dimensional affine Nash manifolds with divisorial corners which are class k semialgebraically diffeomorphic, for k>m^2, are also Nash diffeomorphic., 39 pages