hermitian analogues of hilbert s 17 th problem abstract we pose and discuss several hermitian analogues of hilbert s 17 th problem we survey what is known offer many explicit examples and some proofs and give applications to cr geometry we prove one new algebraic theorem a non negative hermitian symmetric polynomial divides a non zero squared norm if and only if it is a quotient of squared norms we also discuss a new example of putinar scheiderer

explicit solution to the singular discrete time stationary linear filtering problem a closed form solution to the stationary discrete time linear filtering problem is obtained explicitly in terms of the system state space matrices in the limiting singular case where the measurement noise tends to zero simple expressions in closed form are obtained for the kalman gain matrix both for uniform and nonuniform rank systems and the explicit eigenstructure of the kalman filter closed loop matrix is derived the minimum error covariance matrices of the a priori and a posteriori filtered estimates are obtained using this special eigenstructure and a remarkably different behavior of the solution in the minimum and nonminimum phase cases is found

on the analyticity of cr diffeomorphisms in any positive cr dimension and cr codimension we provide a construction of real analytic embedded cr structures which are c infty cr equivalent but are inequivalent holomorphically

convergence of formal cr mappings into strongly pseudoconvex cauchy riemann manifolds it is shown that any formal holomorphic mapping sending a real analytic generic submanifold m subset mathbb c n of finite type into a real analytic strongly pseudoconvex cr submanifold m subset mathbb c n is necessarily convergent as a consequence we obtain a positive answer to the long standing open question of whether formal holomorphic maps sending real analytic strongly pseudoconvex hypersurfaces into each other are convergent

algebraicity of local holomorphisms between real algebraic submanifolds of complex spaces we prove that a germ of a holomorphic map f between c n and c n sending one real algebraic submanifold mc n into another m c n is algebraic provided mcontains no complex analytic discs and m is generic and minimal we also propose an algorithm for finding complex analytic discs in a real submanifold

rigidity of proper holomorphic maps between bounded symmetric domains our first main result gives assumptions guaranteeing that proper holomorphic maps between cartan type i bounded symmetric domains have simple block matrix shape answering positively a question of mok the proof is based on the second main result establishing similar phenomenon for local cr maps between arbitrary boundary components of two bounded symmetric domains of the above type since boundary components other than shilov boundaries are levi degenerate our analysis is based on their 2 nondegeneracy combining levi forms with higher order tensors

a reflection principle for proper holomorphic mappings of strongly pseudoconvex domains and applications this result is proved in 5 the techniques of 12 and 5 involve detailed computat ion of differential invariants on the boundary another paper philosophically related to the above but involving more elementary techniques is 1 in which proper holomorphic maps of b n to b are shown to be automorphisms more recently this result has been proved on strongly pseudoconvex domains in 2 and in 4 in the paper 3 which is the starting point for the present work completely elementary means are used to obtain a reflection principle for suitable maps of b to b k this in turn leads to a proof of the result of 12 assuming only c 2 smoothness to the boundary further detailed information is obtained about the maps even when k n 1 the elementary techniques of 3 may be adapted to use on strictly

a reflection principle with applications to proper holomorphic mappings j l holomorphic mappings from b into b k we then apply this principle to the problem of characterizing the proper holomorphic mappings from b into b k two such mappings f and g are said to be equivalent if there exist holomorphic automorphisms b and p of b and bk respectively such that f pogo49 there are at present three significant papers on the subject the first by alexander 1 proves that i f f is a proper holomorphic mapping of b into b then f is a holomorphic automorphism of b webster i 5 proved that for n 3 i f f is a proper holomorphic mapping of b into b 1 which is c3 then f maps b into a hyperplane in r hence using alexander s result and properties of the holomorphic automorphisms f is equivalent to the mapping zl z2 z zi z2 zn o alexander s example zl z2 z2 zlz2 z 2 shows that webster s result does not hold for n 2 in a recent paper faran 2 has characterized all proper holomorphic mappings f from b 2 to b 3 that satisfy the additional smoothness condition fec3 2 he proves that such mappings are equivalent to one of the four mappings z w z w 0 z w z zw w2 z w r z 2 l zw w2

algebraicity of holomorphic mappings between real algebraic sets in cn 0 introduction 1 holomorphic nondegenera y of real analytic manifolds 1 1 preliminaries on real submanifolds of c n 1 2 holomorphic nondegeneracy and its propagation 1 3 the levi number and essential finiteness 1 4 holomorphic nondegeneracy of real algebraic sets 2 the segre sets of a real analytic cr submanifold 2 1 complexification of m involution and projections 2 2 definition of the segre sets of m at p0 2 3 homogeneous submanifolds of cr dimension 1 2 4 homogeneous submanifolds of arbitrary cr dimension 2 5 proof of theorem 2 2 1 3 algebraic properties of holomorphic mappings between real algebraic sets 3 1 a generalization of theorems 1 and 4 3 2 propagation of algebralcity 3 3 proof of theorem 3 1 2 3 4 proof of theorem 3 1 8 3 5 an example 3 6 proofs of theorems 1 through 4

proper holomorphic mappings from the two ball to the three ball we prove that a proper mapping of the two ball in cn into the three ball which is c2 on the closed two ball is equivalent to one of four normalized polynomial mappings this improves the known result of faran the proof is basic using taylor expansions

sensitivity properties of a class of discrete time lqg controllers with computation delays sensitivity properties of the discrete time lqg controller which utilizes the optimal state predictor to compensate l step computation delays are discussed we obtain multiplicative decompositions of the sensitivity matrices at the input and output of a plant which clarify the effect of computation delays and imperfect observation furthermore these decompositions are used to show that duality between the sensitivity properties at the input and those at the output holds for l 1 and to identify the asymptotic sensitivity matrices obtained by loop transfer recovery procedures

convergence of formal embeddings between real analytic hypersurfaces in codimension one we show that any formal embedding sending a real analytic strongly pseudoconvex hypersurface m c n n 1 into another such hypersurface m cn 1 is convergent

initial monomial invariants of holomorphic maps we study a new biholomorphic invariant of holomorphic maps between domains in different dimensions based on generic initial ideals we start with the standard generic monomial ideals to find invariants for rational maps of spheres and hyperquadrics giving a readily computable invariant in this important case for example the generic initial monomials distinguish all four inequivalent rational proper maps from the two to the three dimensional ball next we associate to each subspace x subset mathcal o u a generic initial monomial subspace which is invariant under biholomorphic transformations and multiplication by nonzero functions the generic initial monomial subspace is a biholomorphic invariant for holomorphic maps if the target automorphism is linear fractional as in the case of automorphisms of spheres or hyperquadrics

rigidity of cr maps between shilov boundaries of bounded symmetric domains our goal is to establish what seems to be the first rigidity result for cr embeddings between shilov boundaries of bounded symmetric domains of higher rank the result states that any such cr embedding is the standard linear embedding up to cr automorphisms our basic assumption extends precisely the well known optimal bound for the rank one case there are no other restrictions on the ranks in particular the difficult case when the target rank is larger than the source rank is also allowed

asymptotic recovery for discrete time systems an asymptotic recovery design procedure is proposed for square discrete time linear time invariant multivariable systems which allows a state feedback design to be approximately recovered by a dynamic output feedback scheme both the case of negligible processing time compared to the sampling interval and of significant processing time are discussed in the former case it is possible to obtain perfect recovery if the plant is minimum phase and has the smallest possible number of zeros at infinity in other cases good recovery is frequently possible new conditions are found which ensure that the return ratio being recovered exhibits good robustness properties

on the c version of the reflection principle for mappings between cr manifolds we prove a smooth version of the classical schwarz reflection principle for cr mappings between an abstract cr manifold m and a generic cr manifold embedded in euclidean complex space as a consequence of our results we settle a conjecture of x huang formulated in 1994

the equivalence problem and rigidity for hypersurfaces embedded into hyperquadrics we consider the class of levi nondegenerate hypersurfaces m in inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i that admit a local cr transversal embedding near a point p e m into a standard nondegenerate hyperquadric in inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 02i with codimension k n n small compared to the cr dimension n of m we show that for hypersurfaces in this class there is a normal form which is closely related to the embedding such that any local equivalence between two hypersurfaces in normal form must be an automorphism of the associated tangent hyperquadric we also show that if the signature of m and that of the standard hyperquadric in inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 03i are the same then the embedding is rigid in the sense that any other embedding must be the original embedding composed with an automorphism of the quadric

holomorphic mappings between hyperquadrics with small signature difference in this paper we study holomorphic mappings sending a hyperquadric of signature ell in bbb c n into a hyperquadric of signature ell in bbb c n we show theorem 1 1 that if the signature difference ell ell is not too large then the mapping can be normalized by automorphisms of the target hyperquadric to a particularly simple form and in particular the image of the mapping is contained in a complex plane of a dimension that depends only on ell and ell and not on the target dimension n we also prove a hopf lemma type result theorem 1 3 for such mappings

bounding the rank of hermitian forms and rigidity for cr mappings of hyperquadrics using green s hyperplane restriction theorem we prove that the rank of a hermitian form on the space of holomorphic polynomials is bounded by a constant depending only on the maximum rank of the form restricted to affine manifolds as an application we prove a rigidity theorem for cr mappings between hyperquadrics in the spirit of the results of baouendi huang and baouendi ebenfelt huang given a real analytic cr mapping of a hyperquadric not equivalent to a sphere to another hyperquadric q a b either the image of the mapping is contained in a complex affine subspace or a is bounded by a constant depending only on b finally we prove a stability result about existence of nontrivial cr mappings of hyperquadrics that is as long as both a and b are sufficiently large and comparable then there exist cr mappings whose image is not contained in a hyperplane the rigidity result also extends when mapping to hyperquadrics in infinite dimensional hilbert space

on the analyticity of cr mappings between nonminimal hypersurfaces let m subset mathbb c 2 be a connected real analytic hypersurface containing a connected complex hypersurface e subset mathbb c 2 and let f colon m to mathbb c 2 be a smooth cr mapping sending m into another real analytic hypersurface m subset mathbb c 2 in this paper we prove that if f does not collapse e to a point and does not collapse m into the image of e and if the levi form of m vanishes to first order along e then f is real analytic in a neighborhood of e in general the corresponding statement is false if the levi form of m vanishes to second order or higher in view of an example due to the author we also show analogous results in higher dimensions provided that the target m satisfies a certain nondegeneracy condition the main ingredient in the proof which seems to be of independent interest is the prolongation of the system defining a cr mapping sending m into m to a pfaffian system on m with singularities along e the nature of the singularity is described by the order of vanishing of the levi form along e

regularity of mappings into classical domains we study the regularity and algebraicity for mappings into classical domains among other things we establish everywhere regularity and algebraicity results for c 2 cr maps into the set of smooth boundary points of a classical domain where the codimension can be arbitrarily large

discrete time lqg ltr technique for systems with time delays loop transfer recovery ltr is applied for lqg controllers with prediction estimators for a class of square minimum phase discrete time linear systems with time delays we consider the class of plants described by the triple a b c whose time delay structure is characterized by the following hypothesis on the plant impulse response cb cab cad 2b 0 det cad 1b 0 when the estimator gain is fixed by recovery technique it is shown that the loop transfer matrix at the plant input is equal to the transfer matrix obtained with the optimal digital regulator having d sample controller delays described in 6 the dual result corresponding to the situation where the feedback gain is fixed by recovery technique is also analysed

on the regularity of cr mappings between cr manifolds of hypersurface type we prove smooth and analytic versions of the classical schwarz reflection principle for transversal cr mappings between two levi nondegenerate cr manifolds of hypersurface type

complete differential system for the mappings of cr manifolds of nondegenerate levi forms letm andn be real analytic c cr manifolds of hypersurface type of dimensions 2m 1 and 2n 1 m n respectively in this paper we show under generic assumptions that a cr mapping f m n satisfies a certain pfaffian system in the jet space which implies the analyticity and rigidity of cr mappings we use a method of prolongation for the tangential cauchy riemann equations the technique is roughly speaking separating the holomorphic derivatives of cr functions from their complex conjugates and applying the tangential cauchy riemann operators and then counting the order of the missing directional derivatives in the holomorphic part the argument of this paper is purely local thus for instance a manifold should be understood as a germ of a manifold at a reference point and mappings are supposed to preserve the reference point first we recall some basic definitions for other definitions and proofs of the facts that we do not present here the readers are refered to jac let m be a differentiable manifold of dimension 2m 1 a cr structure on m is a subbundle v of the complexified tangent bundle tcm having the following properties i each fiber is of complex dimension m ii v v 0 iii v v v integrability given a cr structure v we have levi form

rigidity and holomorphic segre transversality for holomorphic segre maps let mathcal h n and mathcal h mathcal n denote the complexifications of heisenberg hypersurfaces in mathbf c n and mathbf c n respectively we show that non degenerate holomorphic segre mappings from mathcal h n into mathcal h n with n leq2n 2 possess a partial rigidity property as an application we prove that the holomorphic segre non transversality for a holomorphic segre map from mathcal h n into mathcal h n with n leq2n 2 propagates along segre varieties we also give an example showing that this propagation property of holomorphic segre transversality fails when n 2n 2

the lqg ltr procedure for multivariable feedback control design this paper provides a tutorial overview of the lqg ltr design procedure for linear multivariable feedback systems lqg ltr is interpreted as the solution of a specific weighted h2 tradeoff between transfer functions in the frequency domain properties of this solution are examined for both minimum phase and nonminimum phase systems this leads to a formal weight augmentation procedure for the minimum phase case which permits essentially arbitrary specification of system sensitivity functions in terms of the weights while such arbitrary specifications are not possible for nonminimum phase problems a direct relationship between weights and sensitivities is developed for nonminimum phase siso and certain nonminimum phase mimo cases which guides the weight selection process

rigidity for local holomorphic isometric embeddings from mathbb b n into mathbb b n_1 mathbb b n_m up to conformal factors in this article we study local holomorphic isometric embeddings from mathbb b n into mathbb b n 1 mathbb b n m with respect to the normalized bergman metrics up to conformal factors assume that each conformal factor is smooth nash algebraic then each component of the map is a multi valued holomorphic map between complex euclidean spaces by the algebraic extension theorem derived along the lines of mok and mok and ng applying holomorphic continuation and analyzing real analytic subvarieties carefully we show that each component is either a constant map or a proper holomorphic map between balls applying a linearity criterion of huang we conclude the total geodesy of non constant components

embedding strictly pseudoconvex domains into balls every relatively compact strictly pseudocc nvex domain d with c2 boundary in a stein manifold can be embedded as a closed complex submanifold of a finite dimensional ball however for each n 2 there exist bounded strictly pseudoconvex domains d in cn with real analytic boundary such that no proper holomorphic map from d into any finite dimensional ball extends smoothly to d 0 introduction in this paper we study the representations of bounded strictly pseudoconvex domains d c cn if the boundary of d is of class ck k e 2 3 oo then by a theorem of fornaess 8 and khenkin 13 d can be mapped biholomorphically onto the intersection xn q of a bounded strictly convex domain q c cn with ck boundary and a closed complex submanifold x defirxed in a neighborhood of q in cn x intersecting the boundary of q transversally moreover the map f d x n q extends to a holomorphic map on a neighborhood of d the convex domain q depends on d hence a natural question is whether a similar result holds with q replaced by the unit ball b z z1l zn e cn z 2 e zjl2 n that intersects bbn transversally such that d is biholomorphically equivalent to xn e3ff this question has been mentioned by lempert 14 pinduk 20 bedford 2 and others our main result is that the answer to this question is negative in general if d is as above then every biholomorphism of d onto xne3n extends smoothly to d according to 3 however we will show that not all such domains d admit a proper holomorphic map into a finite dimensional ball that is smooth on d theorem 1 1 a similar local result was obtained independently by faran 7 we shall show that the answer to the question q is positive if we allow the intersections of complex submanifolds with strictly convex domains q c cn with real analytic boundaries theorem 1 2 thetheoremsoffornaess 8 andkhenkin 13 onlygiveanqwith smooth boundary received by the editors april 4t 1985 1980 mathematics subject classification primary 32h05 1research supported by a fellowship from the alfred p sloan foundation t 1986 american mathematical society 0002 9947 86 1 00 25 per page

symmetries in cr complexity theory abstract we introduce the hermitian invariant group f of a proper rational map f between the unit ball in complex euclidean space and a generalized ball in a space of typically higher dimension we use properties of the groups to define the crucial new concepts of essential map and the source rank of a map we prove that every finite subgroup of the source automorphism group is the hermitian invariant group of some rational proper map between balls we prove that f is non compact if and only if f is a totally geodesic embedding we show that f contains an n torus if and only if f is equivalent to a monomial map we show that f contains a maximal compact subgroup if and only if f is equivalent to the juxtaposition of tensor powers we also establish a monotonicity result the group after intersecting with the unitary group does not decrease when a tensor product operation is applied to a polynomial proper map we give a necessary condition for f when the target is a generalized ball to contain automorphisms that move the origin

mappings of real algebraic hypersurfaces a holomorphic function h z defined in a neighborhood of a point z0 e cn is algebraic if there are polynomials qj z j 0 k not all identically zero such that qk z h z k qo z 0 a holomorphic mapping is algebraic if all its components are algebraic a smooth real hypersurface m in cn is algebraic if it is contained in the zero set of a nontrivial real valued polynomial in z and z we assume throughout this paper that n 1 in wi webster proved the following celebrated result if m and m are algebraic hypersurfaces in cn with nondegenerate levi forms and if h is a biholomorphism defined in an open neighborhood of m and mapping m into m then h is algebraic previously it had long been known that if m and m are open subsets of spheres then the components of h are in fact rational functions poincare p tanaka t in this paper we go a step further and give a complete characterization of algebraic hypersurfaces in cn such that any holomorphic mapping with nonvanishing jacobian determinant between two such hypersurfaces must be algebraic our main result is stated in theorem 1 below n by a germ at p0 of a holomorphic vector field in c we shall mean a

real hypersurfaces orders of contact and applications a fundamental problem in the modem theory of several complex variables concerns the boundary behavior of the cauchy riemann equations suppose that d is an open domain in cn and that its boundary m is a smooth real submanifold of cn how does the geometry of m influence the function theory on d one approach to this question is the a neumann problem 6 there one studies the inhomogeneous cauchy riemann equations au a where a is a 0 q form with distribution coefficients on the closed domain d u m satisfying the necessary compatibility condition a a 0 the a neumann problem constructs a particular solution u which is orthogonal to the null space of the complex laplacian kohn 10a has solved this problem and has shown that good local regularity properties for u in terms of a follow from so called subelliptic estimates it is natural to seek geometric conditions on m that are necessary and sufficient for these estimates in this paper we will be concerned with a certain geometric condition on m that we feel is the right one for subellipticity although the motivation for the questions discussed here comes from partial differential equations the techniques come from algebraic geometry to see why we state theorems of kohn greiner and catlin concerning subellipticity kohn 1oa has discovered a sufficient condition for a subelliptic estimate near a fixed point p in m in case d is pseudoconvex and is contained in c2 this condition is equivalent to the finiteness of order of tangency of every complex analytic one dimensional manifold with m at p greiner 13 established the necessity of this condition in

on the c regularity of cr mappings of positive codimension abstract the present paper tackles the c regularity problem for cr maps h m m between c smooth cr submanifolds m m embedded in complex spaces of possibly different dimensions for real hypersurfaces m c n 1 and m c n 1 with n n 1 and m strongly pseudoconvex we prove that every cr transversal map of class c n n 1 that is nowhere c on some non empty open subset of m must send this open subset to the set of d angelo infinite points of m as a corollary we obtain that every cr transversal map h m m of class c n n 1 must be c smooth on a dense open subset of m when m is of d angelo finite type another consequence establishes the following boundary regularity result for proper holomorphic maps in positive codimension given c n 1 and c n 1 pseudoconvex domains with smooth boundaries and both of d angelo finite type n n 1 any proper holomorphic map h that extends c n n 1 smoothly up to must be c smooth on a dense open subset of more generally for cr submanifolds m and m of higher codimensions our main result describes the impact of the existence of a nowhere smooth cr map h m m on the cr geometry of m allowing to extend the previously mentioned results in the hypersurface case to any codimension as well as deriving a number of regularity results for cr maps with d angelo infinite type targets

on the third gap for proper holomorphic maps between balls let f be a proper rational map from the complex ball mathbb b n into mathbb b n with n 7 and 3n 1 le n le 4n 7 then f is equivalent to a map g 0 dots 0 where g is a proper holomorphic map from mathbb b n into mathbb b 3n

extended discrete time ltr synthesis of delayed control systems abstract this paper addresses compensation of dealys within a multi input multi output discrete time feedback loop for application to real time distributed control systems the delay compensation algorithm formulated in this paper is an extension of the standard loop transfer recovery ltr procedure from one step prediction to the general case of p step prediction p 1 it is shown that the steady state minimum variance filter gain is the h 2 minimization solution of the relative error between the target sensitivity matrix and the actual sensitivity matrix for p step prediction p 1 this concept forms the basis for synthesis of robust p step delay compensators p 1 the proposed control synthesis procedure for delay compensation is demonstrated via simulation of the flight control system of an advanced aircraft

boundary regularity of proper holomorphic mappings it s main result is formulated in theorem 2 below although the answer to the conjecture in the general case is still unknown it has been shown to hold in many special cases a fairly detailed history of this subject will be contained in 14 one class of positive answers deals with cases in which one knows that f21 and 2 posses fairly large classes of holomorphic automorphisms that extend holomorphically across the boundaries like for instance reinhardt domains see e g h cartan 8 w kaup 16 and st bell 6 in these situations one even often does not need a smoothness assumption on the boundaries of 1 and 22 on the other hand the first result without a priori knowledge on automorphisms which gave c extendability was proved by ch fefferman in 1974 in 15 and says

boundary characterization of holomorphic isometric embeddings between indefinite hyperbolic spaces abstract we provide in this paper a boundary characterization in terms of the boundary cr invariants or pseudo hermitian invariants for holomorphic isometric embeddings between indefinite hyperbolic spaces of general codimensions

bianalytic maps between free spectrahedra linear matrix inequalities lmis i d sum j 1 g a jx j sum j 1 g a j x j succeq 0 play a role in many areas of applications the set of solutions of an lmi is a spectrahedron lmis in dimension free matrix variables model most problems in linear systems engineering and their solution sets are called free spectrahedra free spectrahedra are exactly the free semialgebraic convex sets this paper studies free analytic maps between free spectrahedra and under certain generically valid irreducibility assumptions classifies all those that are bianalytic the foundation of such maps turns out to be a very small class of birational maps we call convexotonic the convexotonic maps in g variables sit in correspondence with g dimensional algebras if two bounded free spectrahedra mathcal d a and mathcal d b meeting our irreducibility assumptions are free bianalytic with map denoted p then p must after possibly an affine linear transform extend to a convexotonic map corresponding to a g dimensional algebra spanned by u i a 1 ldots u i a g for some unitary u furthermore b and ua are unitarily equivalent the article also establishes a positivstellensatz for free analytic functions whose real part is positive semidefinite on a free spectrahedron and proves a representation for a free analytic map from mathcal d a to mathcal d b not necessarily bianalytic another result shows that a function analytic on any radial expansion of a free spectrahedron is approximable by polynomials uniformly on the spectrahedron these theorems are needed for classifying free bianalytic maps

feedback properties of multivariable systems the role and use of the return difference matrix for linear time invariant multivariable feedback systems the feedback properties of plant disturbance attenuation sensor noise response stability margins and sensitivity to plant and sensor variation are quantitatively related to the bode magnitude versus frequency plots of the singular values of the return difference matrix i l and of the associated inverse return difference matrix i l 1 implied fundamental limits of feedback performance are quantitatively described and design tradeoffs are discussed the penalty function in the stochastic linear quadratic gaussian lqg optimal control problem is found to be a weighted sum of the singular values with the weights determined by the quadratic cost and noise intensity matrices this enables systematic tuning of lqg cost and noise matrices so that the resulting optimal return difference and inversereturn difference meet inequality constraints derived from design specifications on feedback properties the theory has been used to synthesize a multivariable automatic controller for the longitudinal dynamics of an advanced fighter aircraft

extending proper holomorphic mappings of positive codimension in this paper we obtain results on holomorphic continuation of proper holomorphic mappings between pseudoconvex domains with real analytic boundaries in complex spaces of different dimensions equivalently we obtain results concerning the analyticity of cauchy riemann mappings between real analytic pseudoconvex hypersurfaces in complex spaces of different dimensions to begin with we recall the corresponding results for mappings of equidimensional domains let d and d be bounded pseudoconvex domains with smooth boundaries in c if the boundaries of d and d are strictly pseudoconvex or more generally of finite type in the sense of d angelo 15 then every proper holomorphic map of d onto d extends smoothly t o according to the results of bell and catlin 7 8 and diederich and forn ess 19 for mappings between strictly pseudoconvex domains this was proved by fefferman 26 and nirenberg webster and yang 37 if the boundaries of the pseudoconvex domains d d c c are real analytic then every proper holomorphic mapping of d onto d extends holomorphically to a neighborhood of 3 according to baouendi and rothschild 2 and diederich and forn ess 21 this reflection principle was first discovered by lewy 34 and pin6uk 38 for mappings between strictly pseudoconvex domains in the case of biholomorphic mappings between weakly pseudoconvex domains with realanalytic boundaries the result follows from the work of baouendi jacobowitz and treves 5 results in this direction were obtained in recent years by several authors see the papers 4 6 18 22 33 47 49 in this paper we are treating the case when the domains d and d have different dimensions to be specific we assume that d e c and d c c n are bounded pseudoconvex domains with real analytic boundaries and n n l in this situation a proper holomorphic map f d d need not be regular at the boundary for instance there exist proper holomorphic maps of balls of different dimensions

bianalytic free maps between spectrahedra and spectraballs abstract linear matrix inequalities lmis are ubiquitous in real algebraic geometry semidefinite programming control theory and signal processing lmis with dimension free matrix unknowns are central to the theories of completely positive maps and operator algebras operator systems and spaces and serve as the paradigm for matrix convex sets the matricial feasibility set of an lmi is called a free spectrahedron in this article the bianalytic maps between a very general class of ball like free spectrahedra examples of which include row or column contractions and tuples of contractions and arbitrary free spectrahedra are characterized and seen to have an elegant algebraic form they are all highly structured rational maps in the case that both the domain and codomain are ball like these bianalytic maps are explicitly determined and the article gives necessary and sufficient conditions for the existence of such a map with a specified value and derivative at a point in particular this result leads to a classification of automorphism groups of ball like free spectrahedra the proofs depend on a novel free nullstellensatz established only after new tools in free analysis are developed and applied to obtain fine detail geometric in nature locally and algebraic in nature globally about the boundary of ball like free spectrahedra

homotopy equivalence for proper holomorphic mappings abstract we introduce several homotopy equivalence relations for proper holomorphic mappings between balls we provide examples showing that the degree of a rational proper mapping between balls in positive codimension is not a homotopy invariant in domain dimension at least 2 we prove that the set of homotopy classes of rational proper mappings from a ball to a higher dimensional ball is finite by contrast when the target dimension is at least twice the domain dimension it is well known that there are uncountably many spherical equivalence classes we generalize this result by proving that an arbitrary homotopy of rational maps whose endpoints are spherically inequivalent must contain uncountably many spherically inequivalent maps we introduce whitney sequences a precise analogue in higher dimensions of the notion of finite blaschke product in one dimension we show that terms in a whitney sequence are homotopic to monomial mappings and we establish an additional result about the target dimensions of such homotopies

maps from the two ball to the three ball in 4 webster proved the following theorem let b z e c 7 2 3 is a proper holomorphic map which is c a up to the boundary of b then f is linear fractional the purpose of this paper is to extend this result to the case n 2 however the theorem is false when n 2 f need not be linear fractional so instead of trying to prove this false theorem we shall instead try to make a list of maps f bz b a which are c 3 up to the boundary of course given such an f we can get another such by composing f with an automorphism of b 2 and an automorphism of b a so we make the

uniqueness of certain polynomials constant on a line abstract we study a question with connections to linear algebra real algebraic geometry combinatorics and complex analysis let p x y be a polynomial of degree d with n positive coefficients and no negative coefficients such that p 1 when x y 1 a sharp estimate d 2 n 3 is known in this paper we study the p for which equality holds we prove some new results about the form of these sharp polynomials using these new results and using two independent computational methods we give a complete classification of these polynomials up to d 17 the question is motivated by the problem of classification of cr maps between spheres in different dimensions