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The paper gives an upper bound for the valence of harmonic polynomials. An example is given to show that this bound is sharp. Interest in harmonic mappings in the complex plane has increased due to the publication of [1] in 1984. Most of this interest has centered on univalent harmonic mappings. It is, however, the purpose of this paper to present the fundamental valence results of complex valued polynomials which are harmonic in the plane. Specifically a harmonic polynomial is a function P(z) = S(z) + T(z) where S(z) and T(z) are analytic polynomials. The degree of P is defined as the larger of the degrees of S and T. Lyzzaik studied the local topology of harmonic mappings in [4]; in particular he put a bound on the local valence of P but did not obtain a global valence bound. In this paper it will be shown that, apart from certain degenerate examples, P(z) is at most n2-valent, where n is the degree of P. An example is exhibited to show this to be the sharp bound. To prove the n2 valence bound we need the following classical result from algebraic geometry; a proof may be found in [2]. Theorem 1 (Bezout's Theorem). Let A and B be relatively prime polynomials in the real variables x and y with real coefficients, and let deg A = n and deg B = m. Then the two algebraic curves A(x, y) = 0 and B(x, y) = 0 have at most mn points in common. To apply Bezout's theorem to the valence problem we restate it in an equivalent form. Theorem 2 (Alternate form of Bezout's Theorem). Let A and B be polynomials in the real variables x and y with real coefficients. If deg A = n and deg B m, then either A and B have at most mn common zeros or have infinitely many common zeros. Proof. If A and B are relatively prime, then, by Bezout's theorem, the polynomial equations A(x, y) = 0 and B(x, y) = 0 have at most mn common zeros. Otherwise A and B are not relatively prime and they have a greatest common factor, say C. Let deg C = q. If C(x, y) = 0 has no solutions, then A and B have at most (n q)(m q) common zeros. But if there exists a point on C(x, y) = 0 with either Oc 74 0 or 4c 04 o, then, by the implicit function theorem, C(x, y) = 0 Received by the editors June 26, 1995 and, in revised form, September 20, 1996 and January 2, 1997. 1991 Mathematics Subject Classification. Primary 30C55. (?)1998 American Mathematical Society

In this paper, the Chow groups of projective hypersurfaces are studied. We will prove that if the degree of the hypersurface is sufficiently high, its Chow group is “small” in the sense that its formal tangent space vanishes. Then, we will give an example in which the formal tangent space is infinite dimensional.

In this paper we prove that in positive characteristics normal embeddings of connected reductive groups are Frobenius split. As a consequence, they have rational singularities and are thus Cohen-Macaulay varieties. As an application, we study the particular case of reductive monoids, which are affine embeddings of their unit group. In particular, we show that the algebra of regular functions of a normal irreducible reductive monoid M has a good filtration for the action of the unit group of M.

Recently N.A'Campo suggested a construction of a link from a generic immersion of a curve into a 2-disk. It is tightly related to the singularity theory. In this paper, we give a simple procedure to draw a diagram of the link from a picture of the curve., 5 pages, 7 figures

In this paper, we prove Vojta's Main Conjecture for split blowups of products of certain elliptic curves with themselves. We then deduce front the conjecture bounds on the average number of rational points lying on curves on these surfaces, and expound upon this connection for abelian surfaces and rational surfaces.

The convolution product is an important tool in the geometric representation theory. Ginzburg constructed the bivariant Chern class operation from a certain convolution algebra of Lagrangian cycles to the convolution algebra of Borel-Moore homology. In this paper we give some remarks on the Ginzburg bivariant Chern classes.

In this present paper, we provide an interpretation of the deformation space of a determinantal Barlow surface via smoothings.

The notion of an w-local functor is used to formulate and prove a theorem which is claimed to encompass Lefschetz's principle in algebraic geometry. In his foundational work, A. Weil formulates Lefschetz's principle as follows: "For a given value of the characteristic p [=zero or a prime], every result involving only a finite number of points and of varieties, which has been proved for some choice of the universal domain [i.e. an algebraically closed field of characteristic p of infinite transcendence degree over the prime field] remains valid without restriction; there is but one algebraic geometry of characteristic p for each value of p, not one algebraic geometry for each universal domain" ([6, p. 306]). The purpose of this paper in applied logic is to give a proof of this statement, a proof which is motivated by the "typical example" with which Weil follows the above statement. Weil states a theorem which he proves for the universal domain of complex numbers (using complex analytic methods); this is followed by an argument proving the theorem for any universal domain of characteristic zero, an argument which involves successive extensions of an isomorphism of finitely-generated subfields of two universal domains. This type of argument is one commonly found in logic and usually called the "back-and-forth" argument. (See [1] for an expository paper on the use of the "back-and-forth" method.) The theorem of ?1 of this paper may be regarded as a formalization of this argument to prove a general result which may be said to include Lefschetz's principle as stated above. In fact our theorem is just a special case of a theorem of Feferman [3] and our only contribution is the observation that this result has relevance for algebraic geometry. The significance-theoretical, if not practical-of our general theorem lies in the fact that the back-and-forth argument, which enables one to transfer a result from one universal domain to another, is done once and for all. Then to apply the general theorem to a particular result of algebraic geometry only requires a check that the particular result falls Received by the editors July 28, 1971. AMS (MOS) subject class flcations (1970). Primary 14A99, 02H15.

1. In discussing the nonlinear approximations of A. G. Vitushkin [2], G. G. Lorentz [1] points out a discrepancy between Vitushkin's definition of the degree of an algebraic polynomial in several variables and the usual definition of algebraic geometry. Lorentz observes that Vitushkin's reliance on a theorem of algebraic geometry makes this deviation unjustified. This paper is written to substantiate Lorentz and begin to fill the resulting gap in Vitushkin's theory. Vitushkin considers, among others, approximating families F, in a real normed linear space L, of the form