## Affine and Quasi-affine Varieties – 2

Restatement: An affine algebraic variety, the set of common zeros of a collection of polynomials, is an irreducible closed subset of ${A_k}^n$ (with the induced topology). An open subset of an affine variety is called a quasi affine algebraic variety.

In this post examples affine and quasi-affine object varieties will be given after an analysis of the relationship between the Z(*) and I(*) operations are clarified.

Definition 2.1 The radical of any ideal, $\tilde {a}$, is defined as

$\sqrt{\tilde {a}}:=\lbrace f\in A, such \; that \; f^r\in {\tilde {a}}\; for \; some \; r>0 \rbrace.$

An ideal that is equal to its radical is called a radical ideal, i.e. if $I=\sqrt I$.

Example 2.1 The radical of the ideal $4\mathbb{Z}$, integer mutilples of $4$, is $2\mathbb{Z}$. The radical ideal of $5\mathbb{Z}$ is $5\mathbb{Z}$.

Definition 2.2 Hilbert’s Nullstellensatz. If $k$ is an algebraically closed field with $\tilde {a}$ an ideal in $A=k[x_1,\dots ,x_n]$ and $f\in A$ a polynomial that vanishes at all points of $Z(\tilde {a})$, then $f^r\in \tilde {a}$ for some integer $r>0$.

Proposition 2.1 Properties of the operations Z(*) and I(*).

1) If $T_1\subseteq T_2$ are subsets of $A$, then $Z(T_1) \supseteq Z(T_2)$.

2) If $Y_1\subseteq Y_2$ are subsets of ${A_k}^n$, then $I(Y_1) \supseteq I(Y_2)$.

3) For any two subsets $Y_1$, $Y_2$ of ${A_k}^n$, we have $I(Y_1\cup Y_2)=I(Y_1)\cap I(Y_2)$.

4) For any ideal $\tilde {a}\subseteq A$, $I(Z(\tilde {a}))=\sqrt{\tilde {a}}$. This is a consequence of Definition 2.2.

5) For any subset $Y\subseteq {A_k}^n$, $Z(I(Y))=\bar{Y}$, the closure of $Y$.

Definition 2.3 Prime ideals. An ideal $I$ of a commutative ring $R$ is prime if it has the following properties:

1) if $a,b\in R$ such that $ab\in I$, then $a\in I$ or $b\in I$,

2) $I$ is not equal to $R$.

Definition 2.4 Maximal ideals. $I$ is a maximal ideal of a ring $R$ if there are no other ideals contained between $I$ and $R$. Every maximal ideal is prime.

Example 2.2 In the ring of integers, $\mathbb{Z}$, the maximal ideals are the principle ideals generated by a prime number.

Corollary 2.1 If $k$ is algebraically closed

1) There is a $1:1$ correspondence $\{points\; in \; {A_k}^n\}\Leftrightarrow \{ maximal \; ideals \; of \; k[x_1, \dots, x_n]\}$.

2) Every ideal $I$ not a subset of $k[x_1,\dots ,x_n]$ has a zero in ${A_k}^n$.

Example 2.3 Let $X=\{a_1, \dots , a_n \} \subset A_k^1$ be a finite algebraic set. $I(X)$ is generated by the function $(x-a_1)\dots (x-a_r)$, and $Z(I(X))=X$ so $Z(*)$ is the inverse of $I(*)$.

Corollary 2.2 An algebraic set $X\subset {A_k}^n$ is an affine variety iff its ideal $I(X)=k[x_1, \dots, x_n]$ is a prime ideal.

Example 2.4 ${A_k}^1$ has a dimension of $1$ as single points are the only irreducible closed subsets of ${A_k}^1$ not equal to ${A_k}^1$. In fact ${A_k}^n$ has a dimension of $n$.

Example 2.5 If $f$ is an irreducible polynomial in $A=k[x_1, \dots , x_n]$, the affine variety $Y=Z(f)$ is called a surface if $n=3$ and a hyper-surface of $n>3$.

Any finite topological space can be decomposed uniquely into finitely many irreducible spaces.

Definition 2.5 A topological space $X$ is said to be Noetherian if it satisfies a descending chain condition for closed subsets. For any sequence $Y_1\supseteq Y_2\supseteq \dots$ of closed subsets there is an integer $r$ such that $Y_r=Y_{r+1}=\dots$. Another way of saying this is that the closed subsets are stationary.

As $k[x_1,\dots ,x_n]$ is a Noetherian ring this implies that any algebraic set is a Noetherian topological space.

Definition 2.6 Inclusion-reversing is defined for two sets $A$ and $B$ with a mapping $\phi: A \rightarrow S$. Then $\phi$ is inclusion-reversing iff for every pair of sets $a_1, a_2 \in A$ such that $a_1 \subseteq a_2 : \phi(a_2) \subseteq \phi (a_1)$.

Corollary 2.3 There is a one-to-one inclusion-reversing correspondence between algebraic sets in ${A_k}^n$ and radical ideals in $A$, given by $Y \mapsto I(Y)$ and $\tilde {a} \mapsto Z(\tilde {a})$.

To summarize the following correspondences between geometric and algebraic forms have been established:

1) points in affine space $\Leftrightarrow$ maximum ideals in a polynomial ring. (Corollary 2.1)

2) affine varieties $\Leftrightarrow$ prime ideals. (Corollary 2.2)

3) algebraic sets $\Leftrightarrow$ radical ideals. (Corollary 2.3)

## Affine Varieties – 1

Algebra is written geometry: geometry is drawn algebra. Sophie Germain.

Commentary on Algebraic Geometry, Robin Hartstone pages 1 to 3.

A field (i.e. displays additive and  multiplicative associativity, commutativity, distributivity, identity and inverses) $k$ is algebraically  closed if every polynomial with coefficients in $k$ has a root in $k$.

Note: a field of real numbers is not algebraically closed as there is a uni-variate polynomial e.g. $x^2+1=0$ that does not have a solution in the real numbers.

An affine $n$-space over $k$, ${A_k }^n$ is a set of all $n$-tuples of elements of $k$. An element of affine $n$-space, $P\in{A_k }^n$ is called a point and for $P=(a_1,\dots,a_n),\: a_i\in k$, then $a_i$ are called the coordinates of $P$.

Let $A=k[x_1,\dots,x_n]$ be the polynomial ring in $n$ variables over $k$. The elements of $A$ can be interpreted as functions from the affine $n$-space to $k$ by defining $f(P)=f(a_1,\dots,a_n),\: f\in A$ and $P\in {A_k }^n$. If $f\in A$ is a polynomial the set of zeros of f are $Z(f):=\{ P\in {A_k }^n|f(P)=0 \}$. Generally, if $T$ is any irreducible subset of $A$ the zero set of $T$ is defined as the common zeros of all the elements of $T$, and called an algebraic set (if $T$ is not irreducible the zero set is called an algebraic variety) i.e. $Z(P):=\{P\in {A_k }^n|f(P)=0 \: for \: all\: f \in T\}$ So an affine algebraic variety is the set of common zeros of a collection of polynomials.

Ideals are subsets of rings i.e. in the integers even numbers are an ideal as it is an additive subgroup of the ring of integers that absorbs multiplication by elements of the ring, i.e. an ideal $I$ in an ring $R$ is closed under addition and multiplication with arbitrary ring elements,

$(I,+)\in(R,+). ~ \forall x\in I, \: \forall r\in R:x.r \in I. ~ \forall x\in I, \: \forall r \in R: r.x \in I.$

An ideal $I$ of a ring $R$ is called principal if there is an element $a\in R$ such that $I=aR=\{ar:r\in R\}$. The ideal is generated by $a$. e.g the ideals $n\mathbb{Z}$ of the ring of integers $\mathbb{Z}$, are all principal as are all the ideals of $Z$. We note that a function $f$ has the same zero set as its powers, so e.g. $Z(x_1)=Z(x_1^2)$ although the ideals are, of course, different. If $\kappa$ is and ideal of $A$ generated by $T$, then $Z(T)=Z(\kappa)$.

A ring of integers and the polynomial ring over a field are both Noetherian. They satisfy the ascending chain condition on ideals, for $I_1\subseteq \dots \subseteq I_{k-1} \subseteq I_k \subseteq I_{k+1} \subseteq \dots$ there exists an $n$ such that $I_n = I_{n+1} = \dots$ i.e. the ascending chain of ideals terminates. $A$ is a Noetherian ring, $\kappa$ is a finite set of generators $f_1, \dots, f_r$. Thus $Z(T)$ can be expressed as the common zero’s of the finite set of polynomials $f_1, \dots, f_r$.

Definition 1.1 A subset $Y$ of ${A_k }^n$ is an algebraic set if there exits a subset $T \subseteq {A_k }^n$ such that $Y=Z(T)$.

Proposition 1.1 The union of two algebraic sets is an algebraic set. The intersection of any family of algebraic sets is an algebraic set. The empty set and the whole space are algebraic sets.

Definition 1.2 A set $X$, along with a collection of subsets $T$ of it, is a topology if the subsets in $T$ obey, 1) the trivial subset $X$ and the empty set $\phi$ are in $T$, 2) whenever sets $A$ and $B$ are in $T$, the so is $A\cap T$, 3) whenever two or more sets are in $T$, then so is $A\cup T$. A set $X$ for which a topology $T$ has been specified is called a topological space. The subsets of $X$ that belong to $T$ are called the open sets of $X$. If $A\in X$ is an open set then its complement ($X\backslash A)$ is called a closed set. Note we can define a topology on a set $X$ by listing all the closed sets, and taking the open sets to be all their complements. The null set $\phi$ is neither open nor closed.

Example 1.1 $X=\lbrace 1,2,3,4 \rbrace$ with subsets $T=\lbrace \phi, \lbrace 1 \rbrace, \lbrace 2,3,4 \rbrace, \lbrace 1,2,3,4 \rbrace \rbrace$ comprise a topology and $X$ is a topological space.

Definition 1.3 We define the Zariski topology on ${A_k }^n$ to be the topology whose closed sets are all the algebraic sets in ${A_k }^n$. Moreover, any subset $X$ of ${A_k }^n$ will have the topology induced by the Zariski topology on ${A_k }^n$. This will be called the Zariski topology on $X$. The Zariski topology is the standard topology in algebraic geometry.

Example 1.2 The Zariski topology of the affine line ${A_k }^1$. Here the point $P\in{A_k }^1$ and $P=(a_1,\dots,a_n)$ with $a_i\in k$. Let $A=k[x_1]$ be the polynomial ring in one variable over $k$. $f(P)=f(a_1,\dots,a_n), \: f\in {A_k }^1.$ $Z(f)=\{P\in {A_k }^1|f(p)=0\}$ and $Z(T)=\{P\in {A_k }^1|f(p)=0\: for all \: f\in T\}$. Every ideal in $A=k[x_1]$ is a principle. Every algebraic set is the set of simple zeros of a single monic polynomial of the form $f(x)=c(x_1-a_1)(x_1-a_2) \dots (x_1-a_n)$, with all $c,a_i\in k$. Then $Z(f)=\{a_1, \dots,a_n\}$. Thus the algebraic sets in ${A_k }^1$ are the finite subsets (with the empty set) and whole space ($f=0$). The open sets are the empty sets and the complements of the finite subsets.

Definition 1.3 A nonempty subset $Y$ of a topological space is irreducible if it cannot be expressed as the union of two proper subsets each of which is closed in $Y$. The empty set is not irreducible.

Definition 1.4 Space $X$ is disconnected if it is the union of closed subsets with the intersections between them being empty. Otherwise the space is called connected.

Example 1.3 ${A_k }^1$ is irreducible.

Definition 1.4 An affine algebraic variety is an irreducible closed subset of ${A_k }^n$. An open subset of the affine algebraic variety is a quasi-affine variety. Using the above definitions we can establish the connection between geometry and algebra by studying the connections between the algebraic sets in ${A_k }^n$ and the ideals in $k[x_1, \dots, x_n]$. The operation $Z(*)$ takes a subset of $k[x_1, \dots ,x_n]$ or an ideal to an algebraic set. With the following definition we establish a two way correspondence.

Definition 1.5 For a subset $X\subset {A_k }^n$ we call the ideal $I(X)$ $I(X):=\lbrace f \in k[x_1, \dots, x_n] ; f(P)=0 \; for \; all \; P \in X \rbrace \subset k[x_1,\dots, x_n]$ Hence we have defined the following two way correspondence; we have an operation $Z(*)$ which maps ideals, subsets, of the polynomial ring in $n$ variables over $k$ to algebraic sets in ${A_k}^n$ and an operation $I(*)$ which maps algebraic sets in ${A_k}^n$ to ideals in the polynomial ring in $n$ variables over $k$.

## Algebraic Geometry – Introduction

Algebraic geometry is the study of geometries that are founded in the algebras of rings, classically the algebra considered is that of the ring of polynomial equations in several variables. We know from algebraic analysis that a single complex polynomial equation in one variable with degree greater than four cannot, in general, be solved. What is of interest is the geometric structures of the sets of solutions.

The following example by Andreas Gathmann explores the geometry of solutions for a single polynomial equation in two variables. Consider the following:

$C_n = \left\lbrace (x,y)\in\mathbb{C}^2;\,\,y^2=(x-1)(x-2)\dots(x-2n)\right\rbrace \subset \mathbb{C}^2,$

where $n\ge1$. If $x$ is selected there is one value for $y$, $0$, if $x\in\lbrace 1,\dots,2n\rbrace$, otherwise there are two values for $y$. The set of equations look like two copies of the complex plain with the zero points $x\in\lbrace 1,\dots,2n\rbrace$ identified on each copy of the plain: the complex plane parameterizes the values for $x$ with the two copies correspond to the two values of $y$, i.e. the two roots of the number $(x-1)(x-2)\dots(x-2n)$.

A complex non-zero number does not have a distinguished first and second root that can be identified as on the first and second copy of the complex plain. However, consider a path:
$x=re^{i\phi},\;\; 0\leq\phi\leq2\pi,\;\; r>0$ around the complex origin, the square root of this $x$ will be defined by $\sqrt{x}=\sqrt{r}e^\frac{i\phi}{2},\;\; 0\leq\phi\leq2\pi,\;\; r>0$ which gives opposite values at $\phi=0$ and $\phi=2\pi$. If in $C_n$ we travel a path around any of the points $1,\dots,2n$ we move from one of the copies of the complex plain to the other.
To visualize this topological space consider the two copies of the complex plain cut along the zero’s $[1,2],\dots,[2n-1,2n]$ of $C_n$ and glued together along these lines, then compactify by adding points at infinity where the parallel lines on $\mathbb{C}$ meet: this results in a compact surface with $n-1$ handles. Such objects are called surfaces of genus $n-1$.

The process by which the surface can be visualised is shown below for $C_3$, which results in a surface of genus $2$. A surface of genus $0$ is a sphere and that of genus $1$ a torus.

Classical algebraic geometry has been extended to any commutative ring with a unit, such as the integers. The geometries of such rings are determined by their algebraic structures. Grothendiek (1960) defined basic geometrical objects and their connections to number theory as schemes to establish the formalism needed to solve problems. Researchers have tried to extend the relationships between algebra and geometry to arbitrary non-commutative rings.

## A Diophantine equation

This post greatly expands the brief proof included in Ian Stewart’s book – Algebraic Number Theory and Fermat’s Last Equation.

This is a proof of the following Diophantine equation result using the ring .

A discussion of the validity, or otherwise, of this approach is included.

Clearly y cannot be even, otherwise 2|LHS but 8|RHS. Therefore y is an odd number.

Factoring in the ring

Any common factor, , of the bracketed terms would divide their sum and difference.

None of these solutions give proper factors (i.e. a proper factor of n is positive integer not equal to 1 or n, and remembering y is odd) of so the bracketed terms must be co-prime.

Most copies of Euclid’s Elements include Chapters 1 to 6 and Chapter 11. Indeed my favourite copy of 1795, is similar and misses out the perhaps most interesting Book 9 which proves the following:

if a cubic number multiplied by a cubic number makes some number this product is a cubic

Propositions 3,4,5,6 are relevant here.

Using this

The concern with this proof should be that it uses the language of factorisation of integers in the ring . In other words is factorisation into irreducibles unique in ? It is true in ?

Fortunately, the answer is yes as is the ring of integers in where factorisation into irreducibles is unique.

## abc Conjecture and Fermat’s last theorem

If the abc conjecture is assumed an asymptotic proof Fermat’s last theorem follows.

An explict form of the abc conjecture (S Laishram, T Shorey, A Baker) yields the following

Using this form of the abc conjecture

The proofs of Fermat’s last theorem for n = 3,4 (Euler/Guass) and n = 5 (Legendre/Dirichlet) are known.

So for this, and many other connections (for example see previous post on Brocard’s problem), the abc conjecture remains as one of the most significant open problems in number theory.

## Brocard’s problem (1897)

Brocard’s problem, stated simply, asks for what numbers does its factorial added to one equate to a square. Ramanujan posed the problem in a different way; 4, 5, 7 as factorials added to one are squares, find other values.

As is the joy of number theory, the problem is easily stated but as yet it remains open.

Computational searches by Berndt and Galway up to 109 have not revealed any other solutions.

However, a proof of the Brocard – Ramanujan problem remains elusive.

An asymptotic proof can be obtained, analogous to an asymptotic proof of Fermat’s last theorem, if a weak form of the ABC conjecture is assumed. What follows is a brief note on this proof.

The weak form of the ABC conjecture states the following

Closely related, and used in this asymptotic proof, is the weak form of Szpiro’s conjecture

Using these the asymptotic proof of Brocard’s problem is as follows

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## The Siddhantasundara of Jnanaraja

Below is a review I wrote about the recent translation of The Siddhantasundara of Jnanaraja, an important text from India dated to 1503 CE.

The Siddhantasundara of Jnanaraja an English translation by Toke Lindegaard Knudsen, pp 351, £58.00, ISBN 013 978-1-4214-1442-3, John Hopkins University Press (2014)

This is a scholarly yet readily readable work of great merit. It is a translation of a 1503 CE text from the Sanskrit poetic forms with its peculiar and unfamiliar astronomical traditions. The difficulties in translating Sanskrit of this period are formidable: in this text the multi-layered complexities have been superbly unpicked with the author’s crystal clear writing giving interpretations in a rigorous yet accessible style. The original poetry is often crafted by the translator into a pleasing English verse.

The author’s introduction to the translation gives a valuable historical and geographical setting for the writings of Jnanaraja; an outline of Indian astronomy (in particular the cosmology based on a geocentric model and epicycles to describe the motion of planets); the cosmic winds that Jnanaraja thought drove the planets. It is noted by the translator that the cosmology and models of the text are contrary to the prevailing Hindu mount Meru centric traditions, yet no disputes of the time are known and on reading I was mindful how the early Hindu traditions were more accommodating than in the near contemporaneous disputes surrounding Galileo Galilae.

Siddhanta is the Sanskrit term for a text that is both theoretical and practical. The Siddhantasundara is itself in three sections: cosmology, mathematical astronomy and mathematics.

Perhaps the least interesting section of this text, to the readers of The Gazette, will be the first with its florid cosmological descriptions:
‘Struck down by the heat of the rays of the sun, the moons ally, the zodiacal signs swiftly diminish.’

The second and third sections of mathematical astronomy and mathematics are fascinating. Modular arithmetic makes an appearance; at least in the translator’s clear expositions of the poetry, trigonometry is referenced and used to good effect as a tool in multifarious astronomical calculations. I noted that Jnanaraja lived in ‘mid southern India’ within striking distance of Kerala where the schools (1300CE to 1600 CE) developed, amongst other techniques, series expansions of trigonometric functions. I should also like to make readers aware that the quality of geometrical diagrams in the exposition of the poetic texts is outstanding, putting many mathematical texts to shame. The use of a hexadecimal counting system is demonstrated, and the use of astronomical numbers hints back to the long traditions of working with large numbers and the different enumerations of the Jains (prior to 200 CE). The use of Heron’s method of root extraction is demonstrated. And many many more diverting mathematical techniques are expounded from the poetry. A section on the beautiful astronomical instruments mentioned in the text is a reminder, if one was needed, that although during the renaissance the construction of scientific instruments was raised to an art form, this was paralleled elsewhere (and often at a much earlier date). Many of these instruments can still be seen on a huge scale at the Jantar Mantar in Jaipur, which I would suggest is a must see spot to rival the Taj Mahal for any readers of The Gazette travelling in India.

Readers who are familiar with the work of J L Heath (A Manual of Greek Mathematics) will be excited to see hints of Greek mathematics, those familiar with the work of O Neugebauer (The Exact Sciences in Antiquity) will welcome the hints of Babylonian techniques and by the historical notes in the introduction about Islamic rule contemporaneous with Jnanaraja’s life time in his home state. As ever, the routes and dates by which identifiable mathematical techniques arrive and are absorbed are notoriously difficult to quantify: however, the richness and diversity of the mathematics known and used by 1503 CE is well demonstrated in this book.

This is a fascinating and well written scholarly text which acts as a welcome reminder that the Indian astronomical tradition is of great mathematical value independent of its parallel applications in Indian astrology.