THE DYNAMICAL MORDELL-LANG PROBLEM FOR NOETHERIAN SPACES
... satisfies the descending chain condition for its closed subsets, i.e., there exists no infinite descending chain of proper closed subsets. Theorem 1.4. Let X be a Noetherian topological space, and let Φ : X −→ X be a continuous function. Then for each x ∈ X and for each closed subset Y of X, the set ...
... satisfies the descending chain condition for its closed subsets, i.e., there exists no infinite descending chain of proper closed subsets. Theorem 1.4. Let X be a Noetherian topological space, and let Φ : X −→ X be a continuous function. Then for each x ∈ X and for each closed subset Y of X, the set ...
Bertini irreducibility theorems over finite fields
... of degree d in Pnk and S = d Sd . If E is a subset of S, the density of E is |E ∩ Sd | µ(E ) := lim |Sd | d→∞ if the limit exists. Theorem (Poonen 2004) Let k be a finite field, and let X be a smooth quasiprojective subscheme of P2n+1 . Then the set of hypersurfaces H of Pnk such k that H ∩ X is smo ...
... of degree d in Pnk and S = d Sd . If E is a subset of S, the density of E is |E ∩ Sd | µ(E ) := lim |Sd | d→∞ if the limit exists. Theorem (Poonen 2004) Let k be a finite field, and let X be a smooth quasiprojective subscheme of P2n+1 . Then the set of hypersurfaces H of Pnk such k that H ∩ X is smo ...
Homology Group - Computer Science, Stony Brook University
... continuous map, ∀σ ∈ M, σ is a simplex, f (σ ) is a simplex. For each simplex, we can add its gravity center, and subdivide the simplex to multiple ones. The resulting complex is called the gravity center subdivision. Theorem Suppose M and N are simplicial complexes embedded in ℝn , f : M → N is a c ...
... continuous map, ∀σ ∈ M, σ is a simplex, f (σ ) is a simplex. For each simplex, we can add its gravity center, and subdivide the simplex to multiple ones. The resulting complex is called the gravity center subdivision. Theorem Suppose M and N are simplicial complexes embedded in ℝn , f : M → N is a c ...
Classical Yang-Baxter Equation and Some Related Algebraic
... C. Bai, L. Guo and X. Ni, Nonabelian generalized Lax pairs, the classical Yang-Baxter equation and PostLie algebras, to appear in Comm. Math. Phys. arXiv:0910.3262. C. Bai, L. Guo and X. Ni, Generalizations of the classical Yang-Baxter equation and O-operators, preprint 2010. Notations: Let g be a L ...
... C. Bai, L. Guo and X. Ni, Nonabelian generalized Lax pairs, the classical Yang-Baxter equation and PostLie algebras, to appear in Comm. Math. Phys. arXiv:0910.3262. C. Bai, L. Guo and X. Ni, Generalizations of the classical Yang-Baxter equation and O-operators, preprint 2010. Notations: Let g be a L ...
Lecture Notes
... RG = {f : G → R| f is a map of sets}. This is a ring, with addition and multiplication defined componentwise. The zero and the identity are the constant maps with value 0, respectively 1. Then GR w Spec(RG ). Proof. This follows by induction from EGA I.3.1.1. However, as it is necessary in the subse ...
... RG = {f : G → R| f is a map of sets}. This is a ring, with addition and multiplication defined componentwise. The zero and the identity are the constant maps with value 0, respectively 1. Then GR w Spec(RG ). Proof. This follows by induction from EGA I.3.1.1. However, as it is necessary in the subse ...
Some definable Galois theory and examples
... K whose field of constants CK is not necessarily algebraically closed, the results are already in Kolchin’s book [8, Chapter VI, Theorem 3]. In all these cases some translation between different languages is required, and the model-theoretic references may be unknown or rather obscure, at least for ...
... K whose field of constants CK is not necessarily algebraically closed, the results are already in Kolchin’s book [8, Chapter VI, Theorem 3]. In all these cases some translation between different languages is required, and the model-theoretic references may be unknown or rather obscure, at least for ...
Every set has its divisor
... Remark:We can think thatθis the virtual element of empty set.The empty set is not really empty! The coefficients of divisor means how much does a element belong to the set.For example,let a and b are two elements in the set X,then 2a-3b is a divisor of X,it means there are double a in the set X and ...
... Remark:We can think thatθis the virtual element of empty set.The empty set is not really empty! The coefficients of divisor means how much does a element belong to the set.For example,let a and b are two elements in the set X,then 2a-3b is a divisor of X,it means there are double a in the set X and ...
Notes on Galois Theory
... ai X i then f σ (X) = σ(ai )X i . Note that f (X) 7→ f σ (X) gives a homomorphism of K[X] → K ′ [X] which is an isomorphism if σ is an isomorphism. The following lemma will help us prove that a splitting field is unique up to isomorphism. Lemma 9: Let K and K ′ be fields and σ : K → K ′ be an isomor ...
... ai X i then f σ (X) = σ(ai )X i . Note that f (X) 7→ f σ (X) gives a homomorphism of K[X] → K ′ [X] which is an isomorphism if σ is an isomorphism. The following lemma will help us prove that a splitting field is unique up to isomorphism. Lemma 9: Let K and K ′ be fields and σ : K → K ′ be an isomor ...
Contents - Harvard Mathematics Department
... instance, intersections work out much more nicely when intersections at the extra “points at infinity” are included. Moreover, when endowed with the complex topology, (complex) projective varieties are compact, unlike all but degenerate affine varieties (i.e. finite sets). It is when defining the no ...
... instance, intersections work out much more nicely when intersections at the extra “points at infinity” are included. Moreover, when endowed with the complex topology, (complex) projective varieties are compact, unlike all but degenerate affine varieties (i.e. finite sets). It is when defining the no ...
The structure of Coh(P1) 1 Coherent sheaves
... We’ve pretty thoroughly explored the story of line bundles, but obviously, not every coherent sheaf is locally free! Here, we’ll briefly explore the structure of skyscraper sheaves (more generally, torsion sheaves, whose stalks are zero-dimensional at all but finitely many points), and then give a g ...
... We’ve pretty thoroughly explored the story of line bundles, but obviously, not every coherent sheaf is locally free! Here, we’ll briefly explore the structure of skyscraper sheaves (more generally, torsion sheaves, whose stalks are zero-dimensional at all but finitely many points), and then give a g ...
Complexity of intersection of real quadrics and topology of
... and Grassmannians are the real ones) unless differently specified. Similarly all homology and cohomology groups are with coefficients in Z2 . 2. Complexity of intersections of real quadrics The aim of this section is to review the numerical bounds that can be derived from the literature for the homo ...
... and Grassmannians are the real ones) unless differently specified. Similarly all homology and cohomology groups are with coefficients in Z2 . 2. Complexity of intersections of real quadrics The aim of this section is to review the numerical bounds that can be derived from the literature for the homo ...
Introducing Algebraic Number Theory
... Let A be a subring of the integral domain B, with B integral over A. In Problems 3–5 we are going to show that A is a field if and only if B is a field. 3. Assume that B is a field, and let a be a nonzero element of A. Then since a−1 ∈ B, there is an equation of the form (a−1 )n + cn−1 (a−1 )n−1 + · · ...
... Let A be a subring of the integral domain B, with B integral over A. In Problems 3–5 we are going to show that A is a field if and only if B is a field. 3. Assume that B is a field, and let a be a nonzero element of A. Then since a−1 ∈ B, there is an equation of the form (a−1 )n + cn−1 (a−1 )n−1 + · · ...
Finite group schemes
... Let G/k be a group scheme over some field k. Let G0 denote the connected component of G that contains e. One expects that G0 is a subgroup scheme of G. This is indeed true. One needs to prove that the image of G0 ×k G0 ⊂ G ×k G under the multiplication map m : G ×k G → G is contained in G0 . We are ...
... Let G/k be a group scheme over some field k. Let G0 denote the connected component of G that contains e. One expects that G0 is a subgroup scheme of G. This is indeed true. One needs to prove that the image of G0 ×k G0 ⊂ G ×k G under the multiplication map m : G ×k G → G is contained in G0 . We are ...
Projective Geometry
... If you have any diagram of points and lines, you can replace every point with coordinates
with the line of coordinates and vice
versa, and you still have a valid diagram.
If you do this to Pappus’ theorem, you get
...
... If you have any diagram of points and lines, you can replace every point with coordinates
Picard Groups of Affine Curves Victor I. Piercey University of Arizona Math 518
... 9.2 in [AM69]). Thus the maximal ideals are invertible. On the other hand, note that these ideals are not principal. There is a geometric reason for this. It comes down to the fact that when this affine curve is embedded in projective space P2k , the result is a curve of genus 1. If all maximal idea ...
... 9.2 in [AM69]). Thus the maximal ideals are invertible. On the other hand, note that these ideals are not principal. There is a geometric reason for this. It comes down to the fact that when this affine curve is embedded in projective space P2k , the result is a curve of genus 1. If all maximal idea ...
Lesson 3
... Divide 2x³ + 3x² - x + 1 by x + 2 When a cubic is divided by a linear expression, the quotient is a quadratic and the remainder, if any, is a constant. Let the quotient by ax² + bx + c Let the remainder be d. 2x³ + 3x² - x + 1 = (x + 2)(ax² + bx + c) + d ...
... Divide 2x³ + 3x² - x + 1 by x + 2 When a cubic is divided by a linear expression, the quotient is a quadratic and the remainder, if any, is a constant. Let the quotient by ax² + bx + c Let the remainder be d. 2x³ + 3x² - x + 1 = (x + 2)(ax² + bx + c) + d ...
here
... followed by more fallow periods. The first active period was in the mid 1960’s, when Tate, himself, proved that for an abelian variety A over a finite field F of characteristic p and a prime number ` 6= p, the natural map: End(A) ⊗ Q` → EndGF (V` (A)), is bijective, where GF is the absolute Galois g ...
... followed by more fallow periods. The first active period was in the mid 1960’s, when Tate, himself, proved that for an abelian variety A over a finite field F of characteristic p and a prime number ` 6= p, the natural map: End(A) ⊗ Q` → EndGF (V` (A)), is bijective, where GF is the absolute Galois g ...
6. Divisors Definition 6.1. We say that a scheme X is regular in
... for some quadratic polynomial g(x). If we assume that the characteristic is not three, then we may complete the cube to get y 2 = x3 + ax + ab, for some a and b ∈ k. Now any two sets of three collinear points are linearly equivalent (since the equation of one line divided by another line is a ration ...
... for some quadratic polynomial g(x). If we assume that the characteristic is not three, then we may complete the cube to get y 2 = x3 + ax + ab, for some a and b ∈ k. Now any two sets of three collinear points are linearly equivalent (since the equation of one line divided by another line is a ration ...
Mathematics Years 7–10 continuum of key ideas
... Solve equations resulting from substitution into formulas Solve word problems using linear equations Solve linear inequalities Solve linear simultaneous equations using algebraic and graphical techniques ...
... Solve equations resulting from substitution into formulas Solve word problems using linear equations Solve linear inequalities Solve linear simultaneous equations using algebraic and graphical techniques ...
Doc - NSW Syllabus
... Solve equations resulting from substitution into formulas Solve word problems using linear equations Solve linear inequalities Solve linear simultaneous equations using algebraic and graphical techniques ...
... Solve equations resulting from substitution into formulas Solve word problems using linear equations Solve linear inequalities Solve linear simultaneous equations using algebraic and graphical techniques ...
Algebraic variety
In mathematics, algebraic varieties (also called varieties) are one of the central objects of study in algebraic geometry. Classically, an algebraic variety was defined to be the set of solutions of a system of polynomial equations, over the real or complex numbers. Modern definitions of an algebraic variety generalize this notion in several different ways, while attempting to preserve the geometric intuition behind the original definition.Conventions regarding the definition of an algebraic variety differ slightly. For example, some authors require that an ""algebraic variety"" is, by definition, irreducible (which means that it is not the union of two smaller sets that are closed in the Zariski topology), while others do not. When the former convention is used, non-irreducible algebraic varieties are called algebraic sets.The notion of variety is similar to that of manifold, the difference being that a variety may have singular points, while a manifold will not. In many languages, both varieties and manifolds are named by the same word.Proven around the year 1800, the fundamental theorem of algebra establishes a link between algebra and geometry by showing that a monic polynomial (an algebraic object) in one variable with complex coefficients is determined by the set of its roots (a geometric object) in the complex plane. Generalizing this result, Hilbert's Nullstellensatz provides a fundamental correspondence between ideals of polynomial rings and algebraic sets. Using the Nullstellensatz and related results, mathematicians have established a strong correspondence between questions on algebraic sets and questions of ring theory. This correspondence is the specificity of algebraic geometry among the other subareas of geometry.