Exemplo n.º 1
0
/**
 * Compute the primitive part and the content of a modular multivariate
 * polynomial e \in Z_p[x_n][x_0, \ldots, x_{n-1}], i.e. e is considered
 * a polynomial in variables x_0, \ldots, x_{n-1} with coefficients being
 * modular polynomials Z_p[x_n]
 * @param e polynomial to operate on
 * @param pp primitive part of @a e, will be computed by this function
 * @param c content (in the sense described above) of @a e, will be
 *        computed by this function
 * @param vars variables x_0, \ldots, x_{n-1}, x_n
 * @param p modulus
 */
void primpart_content(ex& pp, ex& c, ex e, const exvector& vars,
		      const long p)
{
	static const ex ex1(1);
	static const ex ex0(0);
	e = e.expand();
	if (e.is_zero()) {
		pp = ex0;
		c = ex1;
		return;
	}
	exvector rest_vars = vars;
	rest_vars.pop_back();
	ex_collect_t ec;
	collect_vargs(ec, e, rest_vars);

	if (ec.size() == 1) {
		// the input polynomial factorizes into 
		// p_1(x_n) p_2(x_0, \ldots, x_{n-1})
		c = ec.rbegin()->second;
		ec.rbegin()->second = ex1;
		pp = ex_collect_to_ex(ec, rest_vars).expand().smod(numeric(p));
		return;
	}

	// Start from the leading coefficient (which is stored as a last
	// element of the terms array)
	auto i = ec.rbegin();
	ex g = i->second;
	// there are at least two terms, so it's safe to...
	++i;
	while (i != ec.rend() && !g.is_equal(ex1)) {
		g = euclid_gcd(i->second, g, vars.back(), p);
		++i;
	}
	if (g.is_equal(ex1)) {
		pp = e;
		c = ex1;
		return;
	}
	exvector mainvar;
	mainvar.push_back(vars.back());
	for (i = ec.rbegin(); i != ec.rend(); ++i) {
		ex tmp(0);
		if (!divide_in_z_p(i->second, g, tmp, mainvar, p))
			throw std::logic_error(std::string(__func__) +
					": bogus division failure");
		i->second = tmp;
	}

	pp = ex_collect_to_ex(ec, rest_vars).expand().smod(numeric(p));
	c = g;
}
Exemplo n.º 2
0
/** 
 * Exact polynomial division of a, b \in Z_p[x_0, \ldots, x_n]
 * It doesn't check whether the inputs are proper polynomials, so be careful
 * of what you pass in.
 *  
 * @param a  first multivariate polynomial (dividend)
 * @param b  second multivariate polynomial (divisor)
 * @param q  quotient (returned)
 * @param var variables X iterator to first element of vector of symbols
 *
 * @return "true" when exact division succeeds (the quotient is returned in
 *          q), "false" otherwise.
 * @note @a p = 0 means the base ring is Z
 */
bool divide_in_z_p(const ex &a, const ex &b, ex &q, const exvector& vars, const long p)
{
	static const ex _ex1(1);
	if (b.is_zero())
		throw(std::overflow_error("divide_in_z: division by zero"));
	if (b.is_equal(_ex1)) {
		q = a;
		return true;
	}
	if (is_exactly_a<numeric>(a)) {
		if (is_exactly_a<numeric>(b)) {
			// p == 0 means division in Z
			if (p == 0) {
				const numeric tmp = ex_to<numeric>(a/b);
				if (tmp.is_integer()) {
					q = tmp;
					return true;
				} else
					return false;
			} else {
				q = (a*recip(ex_to<numeric>(b), p)).smod(numeric(p));
				return true;
			}
		} else
			return false;
	}
	if (a.is_equal(b)) {
		q = _ex1;
		return true;
	}

	// Main symbol
	const ex &x = vars.back();

	// Compare degrees
	int adeg = a.degree(x), bdeg = b.degree(x);
	if (bdeg > adeg)
		return false;

	// Polynomial long division (recursive)
	ex r = a.expand();
	if (r.is_zero())
		return true;
	int rdeg = adeg;
	ex eb = b.expand();
	ex blcoeff = eb.coeff(x, bdeg);
	exvector v;
	v.reserve(std::max(rdeg - bdeg + 1, 0));
	exvector rest_vars(vars);
	rest_vars.pop_back();
	while (rdeg >= bdeg) {
		ex term, rcoeff = r.coeff(x, rdeg);
		if (!divide_in_z_p(rcoeff, blcoeff, term, rest_vars, p))
			break;
		term = (term*power(x, rdeg - bdeg)).expand();
		v.push_back(term);
		r = (r - term*eb).expand();
		if (p != 0)
			r = r.smod(numeric(p));
		if (r.is_zero()) {
			q = (new add(v))->setflag(status_flags::dynallocated);
			return true;
		}
		rdeg = r.degree(x);
	}
	return false;
}