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236
Task/Cyclotomic-polynomial/C-sharp/cyclotomic-polynomial.cs
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236
Task/Cyclotomic-polynomial/C-sharp/cyclotomic-polynomial.cs
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using System;
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using System.Collections;
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using System.Collections.Generic;
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using System.Linq;
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using IntMap = System.Collections.Generic.Dictionary<int, int>;
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public static class CyclotomicPolynomial
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{
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public static void Main2() {
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Console.WriteLine("Task 1: Cyclotomic polynomials for n <= 30:");
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for (int i = 1; i <= 30; i++) {
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var p = GetCyclotomicPolynomial(i);
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Console.WriteLine($"CP[{i}] = {p.ToString()}");
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}
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Console.WriteLine();
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Console.WriteLine("Task 2: Smallest cyclotomic polynomial with n or -n as a coefficient:");
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for (int i = 1, n = 0; i <= 10; i++) {
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while (true) {
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n++;
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var p = GetCyclotomicPolynomial(n);
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if (p.Any(t => Math.Abs(t.Coefficient) == i)) {
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Console.WriteLine($"CP[{n}] has coefficient with magnitude = {i}");
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n--;
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break;
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}
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}
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}
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}
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private const int MaxFactors = 100_000;
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private const int Algorithm = 2;
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private static readonly Term x = new Term(1, 1);
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private static readonly Dictionary<int, Polynomial> polyCache =
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new Dictionary<int, Polynomial> { [1] = x - 1 };
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private static readonly Dictionary<int, IntMap> factorCache =
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new Dictionary<int, IntMap> { [2] = new IntMap { [2] = 1 } };
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private static Polynomial GetCyclotomicPolynomial(in int n) {
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if (polyCache.TryGetValue(n, out var result)) return result;
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var factors = GetFactors(n);
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if (factors.ContainsKey(n)) { //n is prime
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result = new Polynomial(from exp in ..n select x[exp]);
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} else if (factors.Count == 2 && factors.Contains(2, 1) && factors.Contains(n/2, 1)) { //n = 2p
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result = new Polynomial(from i in ..(n/2) select (IsOdd(i) ? -x : x)[i]);
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} else if (factors.Count == 1 && factors.TryGetValue(2, out int h)) { //n = 2^h
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result = x[1<<(h-1)] + 1;
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} else if (factors.Count == 1 && !factors.ContainsKey(n)) { // n = p^k
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(int p, int k) = factors.First();
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result = new Polynomial(from i in ..p select x[i * (int)Math.Pow(p, k-1)]);
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} else if (factors.Count == 2 && factors.ContainsKey(2)) { //n = 2^h * p^k
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(int p, int k) = factors.First(entry => entry.Key != 2);
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int twoExp = 1 << (factors[2] - 1);
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result = new Polynomial(from i in ..p select (IsOdd(i) ? -x : x)[i * twoExp * (int)Math.Pow(p, k-1)]);
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} else if (factors.ContainsKey(2) && IsOdd(n/2) && n/2 > 1) { // CP(2m)[x] = CP[-m][x], n is odd > 1
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Polynomial cycloDiv2 = GetCyclotomicPolynomial(n/2);
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result = new Polynomial(from term in cycloDiv2 select IsOdd(term.Exponent) ? -term : term);
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#pragma warning disable CS0162
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} else if (Algorithm == 0) {
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var divisors = GetDivisors(n);
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result = x[n] - 1;
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foreach (int d in divisors) result /= GetCyclotomicPolynomial(d);
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} else if (Algorithm == 1) {
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var divisors = GetDivisors(n).ToList();
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int maxDivisor = divisors.Max();
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result = (x[n] - 1) / (x[maxDivisor] - 1);
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foreach (int d in divisors.Where(div => maxDivisor % div == 0)) {
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result /= GetCyclotomicPolynomial(d);
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}
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} else if (Algorithm == 2) {
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int m = 1;
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result = GetCyclotomicPolynomial(m);
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var primes = factors.Keys.ToList();
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primes.Sort();
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foreach (int prime in primes) {
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var cycloM = result;
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result = new Polynomial(from term in cycloM select term.Coefficient * x[term.Exponent * prime]);
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result /= cycloM;
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m *= prime;
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}
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int s = n / m;
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result = new Polynomial(from term in result select term.Coefficient * x[term.Exponent * s]);
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#pragma warning restore CS0162
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} else {
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throw new InvalidOperationException("Invalid algorithm");
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}
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polyCache[n] = result;
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return result;
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}
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private static bool IsOdd(int i) => (i & 1) != 0;
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private static bool Contains(this IntMap map, int key, int value) => map.TryGetValue(key, out int v) && v == value;
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private static int GetOrZero(this IntMap map, int key) => map.TryGetValue(key, out int v) ? v : 0;
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private static IEnumerable<T> Select<T>(this Range r, Func<int, T> f) => Enumerable.Range(r.Start.Value, r.End.Value - r.Start.Value).Select(f);
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private static IntMap GetFactors(in int n) {
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if (factorCache.TryGetValue(n, out var factors)) return factors;
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factors = new IntMap();
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if (!IsOdd(n)) {
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foreach (var entry in GetFactors(n/2)) factors.Add(entry.Key, entry.Value);
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factors[2] = factors.GetOrZero(2) + 1;
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return Cache(n, factors);
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}
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for (int i = 3; i * i <= n; i+=2) {
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if (n % i == 0) {
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foreach (var entry in GetFactors(n/i)) factors.Add(entry.Key, entry.Value);
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factors[i] = factors.GetOrZero(i) + 1;
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return Cache(n, factors);
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}
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}
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factors[n] = 1;
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return Cache(n, factors);
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}
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private static IntMap Cache(int n, IntMap factors) {
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if (n < MaxFactors) factorCache[n] = factors;
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return factors;
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}
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private static IEnumerable<int> GetDivisors(int n) {
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for (int i = 1; i * i <= n; i++) {
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if (n % i == 0) {
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yield return i;
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int div = n / i;
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if (div != i && div != n) yield return div;
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}
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}
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}
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public sealed class Polynomial : IEnumerable<Term>
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{
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public Polynomial() { }
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public Polynomial(params Term[] terms) : this(terms.AsEnumerable()) { }
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public Polynomial(IEnumerable<Term> terms) {
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Terms.AddRange(terms);
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Simplify();
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}
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private List<Term>? terms;
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private List<Term> Terms => terms ??= new List<Term>();
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public int Count => terms?.Count ?? 0;
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public int Degree => Count == 0 ? -1 : Terms[0].Exponent;
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public int LeadingCoefficient => Count == 0 ? 0 : Terms[0].Coefficient;
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public IEnumerator<Term> GetEnumerator() => Terms.GetEnumerator();
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IEnumerator IEnumerable.GetEnumerator() => GetEnumerator();
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public override string ToString() => Count == 0 ? "0" : string.Join(" + ", Terms).Replace("+ -", "- ");
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public static Polynomial operator *(Polynomial p, Term t) => new Polynomial(from s in p select s * t);
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public static Polynomial operator +(Polynomial p, Polynomial q) => new Polynomial(p.Terms.Concat(q.Terms));
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public static Polynomial operator -(Polynomial p, Polynomial q) => new Polynomial(p.Terms.Concat(q.Terms.Select(t => -t)));
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public static Polynomial operator *(Polynomial p, Polynomial q) => new Polynomial(from s in p from t in q select s * t);
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public static Polynomial operator /(Polynomial p, Polynomial q) => p.Divide(q).quotient;
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public (Polynomial quotient, Polynomial remainder) Divide(Polynomial divisor) {
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if (Degree < 0) return (new Polynomial(), this);
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Polynomial quotient = new Polynomial();
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Polynomial remainder = this;
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int lcv = divisor.LeadingCoefficient;
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int dv = divisor.Degree;
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while (remainder.Degree >= divisor.Degree) {
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int lcr = remainder.LeadingCoefficient;
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Term div = new Term(lcr / lcv, remainder.Degree - dv);
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quotient.Terms.Add(div);
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remainder += divisor * -div;
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}
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quotient.Simplify();
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remainder.Simplify();
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return (quotient, remainder);
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}
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private void Simplify() {
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if (Count < 2) return;
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Terms.Sort((a, b) => -a.CompareTo(b));
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for (int i = Terms.Count - 1; i > 0; i--) {
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Term s = Terms[i-1];
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Term t = Terms[i];
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if (t.Exponent == s.Exponent) {
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Terms[i-1] = new Term(s.Coefficient + t.Coefficient, s.Exponent);
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Terms.RemoveAt(i);
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}
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}
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Terms.RemoveAll(t => t.IsZero);
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}
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}
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public readonly struct Term : IEquatable<Term>, IComparable<Term>
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{
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public Term(int coefficient, int exponent = 0) => (Coefficient, Exponent) = (coefficient, exponent);
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public Term this[int exponent] => new Term(Coefficient, exponent); //Using x[exp] because x^exp has low precedence
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public int Coefficient { get; }
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public int Exponent { get; }
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public bool IsZero => Coefficient == 0;
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public static Polynomial operator +(Term left, Term right) => new Polynomial(left, right);
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public static Polynomial operator -(Term left, Term right) => new Polynomial(left, -right);
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public static implicit operator Term(int coefficient) => new Term(coefficient);
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public static Term operator -(Term t) => new Term(-t.Coefficient, t.Exponent);
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public static Term operator *(Term left, Term right) => new Term(left.Coefficient * right.Coefficient, left.Exponent + right.Exponent);
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public static bool operator ==(Term left, Term right) => left.Equals(right);
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public static bool operator !=(Term left, Term right) => !left.Equals(right);
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public static bool operator <(Term left, Term right) => left.CompareTo(right) < 0;
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public static bool operator >(Term left, Term right) => left.CompareTo(right) > 0;
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public static bool operator <=(Term left, Term right) => left.CompareTo(right) <= 0;
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public static bool operator >=(Term left, Term right) => left.CompareTo(right) >= 0;
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public bool Equals(Term other) => Exponent == other.Exponent && Coefficient == other.Coefficient;
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public override bool Equals(object? obj) => obj is Term t && Equals(t);
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public override int GetHashCode() => Coefficient.GetHashCode() * 31 + Exponent.GetHashCode();
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public int CompareTo(Term other) {
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int c = Exponent.CompareTo(other.Exponent);
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if (c != 0) return c;
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return Coefficient.CompareTo(other.Coefficient);
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}
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public override string ToString() => (Coefficient, Exponent) switch {
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(0, _) => "0",
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(_, 0) => $"{Coefficient}",
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(1, 1) => "x",
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(-1, 1) => "-x",
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(_, 1) => $"{Coefficient}x",
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(1, _) => $"x^{Exponent}",
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(-1, _) => $"-x^{Exponent}",
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_ => $"{Coefficient}x^{Exponent}"
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};
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}
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}
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