Example program (generated by Claude)

qqbar_example.c

/*
 * Demonstration of qqbar_sgn_re from FLINT.
 *
 * We construct three sums of nontrivial complex algebraic numbers whose
 * real parts are negative, zero, and positive respectively, using roots
 * of unity via qqbar_root_of_unity(res, p, q) = exp(2*pi*i * p/q).
 *
 * Example 1 (negative):
 *   zeta_3 + zeta_3^2  =  e^(2pi*i/3) + e^(2pi*i*2/3)
 *   Each summand is a primitive cube root of unity with Re = -1/2.
 *   Sum has Re = -1.
 *
 * Example 2 (zero):
 *   zeta_8 + zeta_8^3  =  e^(2pi*i/8) + e^(2pi*i*3/8)
 *   Re(zeta_8)  =  cos(pi/4)  =  +sqrt(2)/2
 *   Re(zeta_8^3) = cos(3*pi/4) = -sqrt(2)/2
 *   Sum has Re = 0.
 *
 * Example 3 (positive):
 *   zeta_5 + zeta_5^4  =  e^(2pi*i/5) + e^(2pi*i*4/5)
 *   These are complex conjugates, so Im cancels.
 *   Re = 2*cos(2*pi/5) = (sqrt(5)-1)/2 > 0.
 */
 
#include <stdio.h>
#include <flint/qqbar.h>
 
static void print_sgn(int s)
{
    if      (s < 0) printf("negative (-1)\n");
    else if (s > 0) printf("positive (+1)\n");
    else            printf("zero (0)\n");
}
 
int main(void)
{
    qqbar_t a, b, sum;
 
    qqbar_init(a);
    qqbar_init(b);
    qqbar_init(sum);
 
    /* --- Example 1: Re(zeta_3 + zeta_3^2) = -1 (negative) --- */
    printf("Example 1: zeta_3 + zeta_3^2,  where zeta_3 = exp(2*pi*i/3)\n");
    printf("  zeta_3   = "); qqbar_root_of_unity(a, 1, 3); qqbar_printn(a, 6); printf("\n");
    printf("  zeta_3^2 = "); qqbar_root_of_unity(b, 2, 3); qqbar_printn(b, 6); printf("\n");
    qqbar_add(sum, a, b);
    printf("  sum      = "); qqbar_printn(sum, 6); printf("\n");
    printf("  sgn_re   = ");
    print_sgn(qqbar_sgn_re(sum));
    printf("\n");
 
    /* --- Example 2: Re(zeta_8 + zeta_8^3) = 0 (zero) --- */
    printf("Example 2: zeta_8 + zeta_8^3,  where zeta_8 = exp(2*pi*i/8)\n");
    printf("  zeta_8   = "); qqbar_root_of_unity(a, 1, 8); qqbar_printn(a, 6); printf("\n");
    printf("  zeta_8^3 = "); qqbar_root_of_unity(b, 3, 8); qqbar_printn(b, 6); printf("\n");
    qqbar_add(sum, a, b);
    printf("  sum      = "); qqbar_printn(sum, 6); printf("\n");
    printf("  sgn_re   = ");
    print_sgn(qqbar_sgn_re(sum));
    printf("\n");
 
    /* --- Example 3: Re(zeta_5 + zeta_5^4) = (sqrt(5)-1)/2 > 0 (positive) --- */
    printf("Example 3: zeta_5 + zeta_5^4,  where zeta_5 = exp(2*pi*i/5)\n");
    printf("  zeta_5   = "); qqbar_root_of_unity(a, 1, 5); qqbar_printn(a, 6); printf("\n");
    printf("  zeta_5^4 = "); qqbar_root_of_unity(b, 4, 5); qqbar_printn(b, 6); printf("\n");
    qqbar_add(sum, a, b);
    printf("  sum      = "); qqbar_printn(sum, 6); printf("\n");
    printf("  sgn_re   = ");
    print_sgn(qqbar_sgn_re(sum));
    printf("\n");
 
    qqbar_clear(a);
    qqbar_clear(b);
    qqbar_clear(sum);
 
    return 0;
}