static void precompute_dinv_p(fmpz *list, long M, long d, long p, long N) { fmpz_one(list + 0); if (M >= p) { fmpz_t P, PN; long r; fmpz_init_set_ui(P, p); fmpz_init(PN); fmpz_pow_ui(PN, P, N); fmpz_set_ui(list + 1, d); _padic_inv(list + 1, list + 1, P, N); for (r = 2; r <= M / p; r++) { fmpz_mul(list + r, list + (r - 1), list + 1); fmpz_mod(list + r, list + r, PN); } fmpz_clear(P); fmpz_clear(PN); } }
static void _padic_log_bsplit(fmpz_t z, const fmpz_t y, long v, const fmpz_t p, long N) { fmpz_t P, B, T; long n; if (fmpz_fits_si(p)) n = _padic_log_bound(v, N, fmpz_get_si(p)); else n = (N - 1) / v; n = FLINT_MAX(n, 2); fmpz_init(P); fmpz_init(B); fmpz_init(T); _padic_log_bsplit_series(P, B, T, y, 1, n); n = fmpz_remove(B, B, p); fmpz_pow_ui(P, p, n); fmpz_divexact(T, T, P); _padic_inv(B, B, p, N); fmpz_mul(z, T, B); fmpz_clear(P); fmpz_clear(B); fmpz_clear(T); }
static void entry(fmpz_t rop_u, long *rop_v, const long *u, const long *v, const fmpz *a, const fmpz *dinv, const fmpz **mu, long M, const long **C, const long *lenC, long n, long d, long p, long N, long N2) { const long ku = diagfrob_k(u, n, d); const long kv = diagfrob_k(v, n, d); fmpz_t f, g, P; fmpz_init(f); fmpz_init(g); fmpz_init_set_ui(P, p); /* Compute $g := (u'-1)! \alpha_{u+1,v+1}$ to precision $N2$. */ fmpz_fac_ui(f, ku - 1); alpha(g, u, v, a, dinv, mu, M, C, lenC, n, d, p, N2); fmpz_mul(g, f, g); /* Compute $f := (-1)^{u'+v'} (v'-1)!$ exactly. */ fmpz_fac_ui(f, kv - 1); if ((ku + kv) % 2 != 0) { fmpz_neg(f, f); } /* Set rop to the product of $f$ and $g^{-1} mod $p^N$. */ *rop_v = fmpz_remove(f, f, P) + n - fmpz_remove(g, g, P); if (*rop_v >= N) { fmpz_zero(rop_u); *rop_v = 0; } else { _padic_inv(g, g, P, N - *rop_v); fmpz_mul(rop_u, f, g); fmpz_pow_ui(f, P, N - *rop_v); fmpz_mod(rop_u, rop_u, f); } fmpz_clear(f); fmpz_clear(g); fmpz_clear(P); }
static void precompute_dinv_2(fmpz *list, long M, long d, long N) { fmpz_one(list + 0); if (M >= 2) { const fmpz_t P = {2L}; long r; fmpz_set_ui(list + 1, d); _padic_inv(list + 1, list + 1, P, N); for (r = 2; r <= M / 2; r++) { fmpz_mul(list + r, list + (r - 1), list + 1); fmpz_fdiv_r_2exp(list + r, list + r, N); } } }
static void _qadic_exp_bsplit(fmpz *y, const fmpz *x, slong v, slong len, const fmpz *a, const slong *j, slong lena, const fmpz_t p, slong N) { const slong d = j[lena - 1]; const slong n = _padic_exp_bound(v, N, p); if (n == 1) { fmpz_one(y + 0); _fmpz_vec_zero(y + 1, d - 1); } else { fmpz *P, *T; fmpz_t Q, R; slong f; P = _fmpz_vec_init(2*d - 1); T = _fmpz_vec_init(2*d - 1); fmpz_init(Q); fmpz_init(R); _qadic_exp_bsplit_series(P, Q, T, x, len, 1, n, a, j, lena); fmpz_add(T + 0, T + 0, Q); /* (T,Q) := (T,Q) + 1 */ /* Note exp(x) is a unit so val(T) == val(Q). */ f = fmpz_remove(Q, Q, p); fmpz_pow_ui(R, p, f); _fmpz_vec_scalar_divexact_fmpz(T, T, d, R); _padic_inv(Q, Q, p, N); _fmpz_vec_scalar_mul_fmpz(y, T, d, Q); _fmpz_vec_clear(P, 2*d - 1); _fmpz_vec_clear(T, 2*d - 1); fmpz_clear(Q); fmpz_clear(R); } }
void frob(const mpoly_t P, const ctx_t ctxFracQt, const qadic_t t1, const qadic_ctx_t Qq, prec_t *prec, const prec_t *prec_in, int verbose) { const padic_ctx_struct *Qp = &Qq->pctx; const fmpz *p = Qp->p; const long a = qadic_ctx_degree(Qq); const long n = P->n - 1; const long d = mpoly_degree(P, -1, ctxFracQt); const long b = gmc_basis_size(n, d); long i, j, k; /* Diagonal fibre */ padic_mat_t F0; /* Gauss--Manin Connection */ mat_t M; mon_t *bR, *bC; fmpz_poly_t r; /* Local solution */ fmpz_poly_mat_t C, Cinv; long vC, vCinv; /* Frobenius */ fmpz_poly_mat_t F; long vF; fmpz_poly_mat_t F1; long vF1; fmpz_poly_t cp; clock_t c0, c1; double c; if (verbose) { printf("Input:\n"); printf(" P = "), mpoly_print(P, ctxFracQt), printf("\n"); printf(" p = "), fmpz_print(p), printf("\n"); printf(" t1 = "), qadic_print_pretty(t1, Qq), printf("\n"); printf("\n"); fflush(stdout); } /* Step 1 {M, r} *********************************************************/ c0 = clock(); mat_init(M, b, b, ctxFracQt); fmpz_poly_init(r); gmc_compute(M, &bR, &bC, P, ctxFracQt); { fmpz_poly_t t; fmpz_poly_init(t); fmpz_poly_set_ui(r, 1); for (i = 0; i < M->m; i++) for (j = 0; j < M->n; j++) { fmpz_poly_lcm(t, r, fmpz_poly_q_denref( (fmpz_poly_q_struct *) mat_entry(M, i, j, ctxFracQt))); fmpz_poly_swap(r, t); } fmpz_poly_clear(t); } c1 = clock(); c = (double) (c1 - c0) / CLOCKS_PER_SEC; if (verbose) { printf("Gauss-Manin connection:\n"); printf(" r(t) = "), fmpz_poly_print_pretty(r, "t"), printf("\n"); printf(" Time = %f\n", c); printf("\n"); fflush(stdout); } { qadic_t t; qadic_init2(t, 1); fmpz_poly_evaluate_qadic(t, r, t1, Qq); if (qadic_is_zero(t)) { printf("Exception (deformation_frob).\n"); printf("The resultant r evaluates to zero (mod p) at t1.\n"); abort(); } qadic_clear(t); } /* Precisions ************************************************************/ if (prec_in != NULL) { *prec = *prec_in; } else { deformation_precisions(prec, p, a, n, d, fmpz_poly_degree(r)); } if (verbose) { printf("Precisions:\n"); printf(" N0 = %ld\n", prec->N0); printf(" N1 = %ld\n", prec->N1); printf(" N2 = %ld\n", prec->N2); printf(" N3 = %ld\n", prec->N3); printf(" N3i = %ld\n", prec->N3i); printf(" N3w = %ld\n", prec->N3w); printf(" N3iw = %ld\n", prec->N3iw); printf(" N4 = %ld\n", prec->N4); printf(" m = %ld\n", prec->m); printf(" K = %ld\n", prec->K); printf(" r = %ld\n", prec->r); printf(" s = %ld\n", prec->s); printf("\n"); fflush(stdout); } /* Initialisation ********************************************************/ padic_mat_init2(F0, b, b, prec->N4); fmpz_poly_mat_init(C, b, b); fmpz_poly_mat_init(Cinv, b, b); fmpz_poly_mat_init(F, b, b); vF = 0; fmpz_poly_mat_init(F1, b, b); vF1 = 0; fmpz_poly_init(cp); /* Step 2 {F0} ***********************************************************/ { padic_ctx_t pctx_F0; fmpz *t; padic_ctx_init(pctx_F0, p, FLINT_MIN(prec->N4 - 10, 0), prec->N4, PADIC_VAL_UNIT); t = _fmpz_vec_init(n + 1); c0 = clock(); mpoly_diagonal_fibre(t, P, ctxFracQt); diagfrob(F0, t, n, d, prec->N4, pctx_F0, 0); padic_mat_transpose(F0, F0); c1 = clock(); c = (double) (c1 - c0) / CLOCKS_PER_SEC; if (verbose) { printf("Diagonal fibre:\n"); printf(" P(0) = {"), _fmpz_vec_print(t, n + 1), printf("}\n"); printf(" Time = %f\n", c); printf("\n"); fflush(stdout); } _fmpz_vec_clear(t, n + 1); padic_ctx_clear(pctx_F0); } /* Step 3 {C, Cinv} ******************************************************/ /* Compute C as a matrix over Z_p[[t]]. A is the same but as a series of matrices over Z_p. Mt is the matrix -M^t, and Cinv is C^{-1}^t, the local solution of the differential equation replacing M by Mt. */ c0 = clock(); { const long K = prec->K; padic_mat_struct *A; gmde_solve(&A, K, p, prec->N3, prec->N3w, M, ctxFracQt); gmde_convert_soln(C, &vC, A, K, p); for(i = 0; i < K; i++) padic_mat_clear(A + i); free(A); } c1 = clock(); c = (double) (c1 - c0) / CLOCKS_PER_SEC; if (verbose) { printf("Local solution:\n"); printf(" Time for C = %f\n", c); fflush(stdout); } c0 = clock(); { const long K = (prec->K + (*p) - 1) / (*p); mat_t Mt; padic_mat_struct *Ainv; mat_init(Mt, b, b, ctxFracQt); mat_transpose(Mt, M, ctxFracQt); mat_neg(Mt, Mt, ctxFracQt); gmde_solve(&Ainv, K, p, prec->N3i, prec->N3iw, Mt, ctxFracQt); gmde_convert_soln(Cinv, &vCinv, Ainv, K, p); fmpz_poly_mat_transpose(Cinv, Cinv); fmpz_poly_mat_compose_pow(Cinv, Cinv, *p); for(i = 0; i < K; i++) padic_mat_clear(Ainv + i); free(Ainv); mat_clear(Mt, ctxFracQt); } c1 = clock(); c = (double) (c1 - c0) / CLOCKS_PER_SEC; if (verbose) { printf(" Time for C^{-1} = %f\n", c); printf("\n"); fflush(stdout); } /* Step 4 {F(t) := C(t) F(0) C(t^p)^{-1}} ********************************/ /* Computes the product C(t) F(0) C(t^p)^{-1} modulo (p^{N_2}, t^K). This is done by first computing the unit part of the product exactly over the integers modulo t^K. */ c0 = clock(); { fmpz_t pN; fmpz_poly_mat_t T; fmpz_init(pN); fmpz_poly_mat_init(T, b, b); for (i = 0; i < b; i++) { /* Find the unique k s.t. F0(i,k) is non-zero */ for (k = 0; k < b; k++) if (!fmpz_is_zero(padic_mat_entry(F0, i, k))) break; if (k == b) { printf("Exception (frob). F0 is singular.\n\n"); abort(); } for (j = 0; j < b; j++) { fmpz_poly_scalar_mul_fmpz(fmpz_poly_mat_entry(T, i, j), fmpz_poly_mat_entry(Cinv, k, j), padic_mat_entry(F0, i, k)); } } fmpz_poly_mat_mul(F, C, T); fmpz_poly_mat_truncate(F, prec->K); vF = vC + padic_mat_val(F0) + vCinv; /* Canonicalise (F, vF) */ { long v = fmpz_poly_mat_ord_p(F, p); if (v == LONG_MAX) { printf("ERROR (deformation_frob). F(t) == 0.\n"); abort(); } else if (v > 0) { fmpz_pow_ui(pN, p, v); fmpz_poly_mat_scalar_divexact_fmpz(F, F, pN); vF = vF + v; } } /* Reduce (F, vF) modulo p^{N2} */ fmpz_pow_ui(pN, p, prec->N2 - vF); fmpz_poly_mat_scalar_mod_fmpz(F, F, pN); fmpz_clear(pN); fmpz_poly_mat_clear(T); } c1 = clock(); c = (double) (c1 - c0) / CLOCKS_PER_SEC; if (verbose) { printf("Matrix for F(t):\n"); printf(" Time = %f\n", c); printf("\n"); fflush(stdout); } /* Step 5 {G = r(t)^m F(t)} **********************************************/ c0 = clock(); { fmpz_t pN; fmpz_poly_t t; fmpz_init(pN); fmpz_poly_init(t); fmpz_pow_ui(pN, p, prec->N2 - vF); /* Compute r(t)^m mod p^{N2-vF} */ if (prec->denR == NULL) { fmpz_mod_poly_t _t; fmpz_mod_poly_init(_t, pN); fmpz_mod_poly_set_fmpz_poly(_t, r); fmpz_mod_poly_pow(_t, _t, prec->m); fmpz_mod_poly_get_fmpz_poly(t, _t); fmpz_mod_poly_clear(_t); } else { /* TODO: We don't really need a copy */ fmpz_poly_set(t, prec->denR); } fmpz_poly_mat_scalar_mul_fmpz_poly(F, F, t); fmpz_poly_mat_scalar_mod_fmpz(F, F, pN); /* TODO: This should not be necessary? */ fmpz_poly_mat_truncate(F, prec->K); fmpz_clear(pN); fmpz_poly_clear(t); } c1 = clock(); c = (double) (c1 - c0) / CLOCKS_PER_SEC; if (verbose) { printf("Analytic continuation:\n"); printf(" Time = %f\n", c); printf("\n"); fflush(stdout); } /* Steps 6 and 7 *********************************************************/ if (a == 1) { /* Step 6 {F(1) = r(t_1)^{-m} G(t_1)} ********************************/ c0 = clock(); { const long N = prec->N2 - vF; fmpz_t f, g, t, pN; fmpz_init(f); fmpz_init(g); fmpz_init(t); fmpz_init(pN); fmpz_pow_ui(pN, p, N); /* f := \hat{t_1}, g := r(\hat{t_1})^{-m} */ _padic_teichmuller(f, t1->coeffs + 0, p, N); if (prec->denR == NULL) { _fmpz_mod_poly_evaluate_fmpz(g, r->coeffs, r->length, f, pN); fmpz_powm_ui(t, g, prec->m, pN); } else { _fmpz_mod_poly_evaluate_fmpz(t, prec->denR->coeffs, prec->denR->length, f, pN); } _padic_inv(g, t, p, N); /* F1 := g G(\hat{t_1}) */ for (i = 0; i < b; i++) for (j = 0; j < b; j++) { const fmpz_poly_struct *poly = fmpz_poly_mat_entry(F, i, j); const long len = poly->length; if (len == 0) { fmpz_poly_zero(fmpz_poly_mat_entry(F1, i, j)); } else { fmpz_poly_fit_length(fmpz_poly_mat_entry(F1, i, j), 1); _fmpz_mod_poly_evaluate_fmpz(t, poly->coeffs, len, f, pN); fmpz_mul(fmpz_poly_mat_entry(F1, i, j)->coeffs + 0, g, t); fmpz_mod(fmpz_poly_mat_entry(F1, i, j)->coeffs + 0, fmpz_poly_mat_entry(F1, i, j)->coeffs + 0, pN); _fmpz_poly_set_length(fmpz_poly_mat_entry(F1, i, j), 1); _fmpz_poly_normalise(fmpz_poly_mat_entry(F1, i, j)); } } vF1 = vF; fmpz_poly_mat_canonicalise(F1, &vF1, p); fmpz_clear(f); fmpz_clear(g); fmpz_clear(t); fmpz_clear(pN); } c1 = clock(); c = (double) (c1 - c0) / CLOCKS_PER_SEC; if (verbose) { printf("Evaluation:\n"); printf(" Time = %f\n", c); printf("\n"); fflush(stdout); } } else { /* Step 6 {F(1) = r(t_1)^{-m} G(t_1)} ********************************/ c0 = clock(); { const long N = prec->N2 - vF; fmpz_t pN; fmpz *f, *g, *t; fmpz_init(pN); f = _fmpz_vec_init(a); g = _fmpz_vec_init(2 * a - 1); t = _fmpz_vec_init(2 * a - 1); fmpz_pow_ui(pN, p, N); /* f := \hat{t_1}, g := r(\hat{t_1})^{-m} */ _qadic_teichmuller(f, t1->coeffs, t1->length, Qq->a, Qq->j, Qq->len, p, N); if (prec->denR == NULL) { fmpz_t e; fmpz_init_set_ui(e, prec->m); _fmpz_mod_poly_compose_smod(g, r->coeffs, r->length, f, a, Qq->a, Qq->j, Qq->len, pN); _qadic_pow(t, g, a, e, Qq->a, Qq->j, Qq->len, pN); fmpz_clear(e); } else { _fmpz_mod_poly_reduce(prec->denR->coeffs, prec->denR->length, Qq->a, Qq->j, Qq->len, pN); _fmpz_poly_normalise(prec->denR); _fmpz_mod_poly_compose_smod(t, prec->denR->coeffs, prec->denR->length, f, a, Qq->a, Qq->j, Qq->len, pN); } _qadic_inv(g, t, a, Qq->a, Qq->j, Qq->len, p, N); /* F1 := g G(\hat{t_1}) */ for (i = 0; i < b; i++) for (j = 0; j < b; j++) { const fmpz_poly_struct *poly = fmpz_poly_mat_entry(F, i, j); const long len = poly->length; fmpz_poly_struct *poly2 = fmpz_poly_mat_entry(F1, i, j); if (len == 0) { fmpz_poly_zero(poly2); } else { _fmpz_mod_poly_compose_smod(t, poly->coeffs, len, f, a, Qq->a, Qq->j, Qq->len, pN); fmpz_poly_fit_length(poly2, 2 * a - 1); _fmpz_poly_mul(poly2->coeffs, g, a, t, a); _fmpz_mod_poly_reduce(poly2->coeffs, 2 * a - 1, Qq->a, Qq->j, Qq->len, pN); _fmpz_poly_set_length(poly2, a); _fmpz_poly_normalise(poly2); } } /* Now the matrix for p^{-1} F_p at t=t_1 is (F1, vF1). */ vF1 = vF; fmpz_poly_mat_canonicalise(F1, &vF1, p); fmpz_clear(pN); _fmpz_vec_clear(f, a); _fmpz_vec_clear(g, 2 * a - 1); _fmpz_vec_clear(t, 2 * a - 1); } c1 = clock(); c = (double) (c1 - c0) / CLOCKS_PER_SEC; if (verbose) { printf("Evaluation:\n"); printf(" Time = %f\n", c); printf("\n"); fflush(stdout); } /* Step 7 {Norm} *****************************************************/ /* Computes the matrix for $q^{-1} F_q$ at $t = t_1$ as the product $F \sigma(F) \dotsm \sigma^{a-1}(F)$ up appropriate transpositions because our convention of columns vs rows is the opposite of that used by Gerkmann. Note that, in any case, transpositions do not affect the characteristic polynomial. */ c0 = clock(); { const long N = prec->N1 - a * vF1; fmpz_t pN; fmpz_poly_mat_t T; fmpz_init(pN); fmpz_poly_mat_init(T, b, b); fmpz_pow_ui(pN, p, N); fmpz_poly_mat_frobenius(T, F1, 1, p, N, Qq); _qadic_mat_mul(F1, F1, T, pN, Qq); for (i = 2; i < a; i++) { fmpz_poly_mat_frobenius(T, T, 1, p, N, Qq); _qadic_mat_mul(F1, F1, T, pN, Qq); } vF1 = a * vF1; fmpz_poly_mat_canonicalise(F1, &vF1, p); fmpz_clear(pN); fmpz_poly_mat_clear(T); } c1 = clock(); c = (double) (c1 - c0) / CLOCKS_PER_SEC; if (verbose) { printf("Norm:\n"); printf(" Time = %f\n", c); printf("\n"); fflush(stdout); } } /* Step 8 {Reverse characteristic polynomial} ****************************/ c0 = clock(); deformation_revcharpoly(cp, F1, vF1, n, d, prec->N0, prec->r, prec->s, Qq); c1 = clock(); c = (double) (c1 - c0) / CLOCKS_PER_SEC; if (verbose) { printf("Reverse characteristic polynomial:\n"); printf(" p(T) = "), fmpz_poly_print_pretty(cp, "T"), printf("\n"); printf(" Time = %f\n", c); printf("\n"); fflush(stdout); } /* Clean up **************************************************************/ padic_mat_clear(F0); mat_clear(M, ctxFracQt); free(bR); free(bC); fmpz_poly_clear(r); fmpz_poly_mat_clear(C); fmpz_poly_mat_clear(Cinv); fmpz_poly_mat_clear(F); fmpz_poly_mat_clear(F1); fmpz_poly_clear(cp); }
void _qadic_norm_resultant(fmpz_t rop, const fmpz *op, slong len, const fmpz *a, const slong *j, slong lena, const fmpz_t p, slong N) { const slong d = j[lena - 1]; fmpz_t pN; fmpz_init(pN); fmpz_pow_ui(pN, p, N); if (len == 1) { fmpz_powm_ui(rop, op + 0, d, pN); } else /* len >= 2 */ { { const slong n = d + len - 1; slong i, k; fmpz *M; M = flint_calloc(n * n, sizeof(fmpz)); for (k = 0; k < len-1; k++) { for (i = 0; i < lena; i++) { M[k * n + k + (d - j[i])] = a[i]; } } for (k = 0; k < d; k++) { for (i = 0; i < len; i++) { M[(len-1 + k) * n + k + (len-1 - i)] = op[i]; } } _fmpz_mod_mat_det(rop, M, n, pN); flint_free(M); } /* XXX: This part of the code is currently untested as the Conway polynomials used for the extension Qq/Qp are monic. */ if (!fmpz_is_one(a + (lena - 1))) { fmpz_t f; fmpz_init(f); fmpz_powm_ui(f, a + (lena - 1), len - 1, pN); _padic_inv(f, f, p, N); fmpz_mul(rop, f, rop); fmpz_mod(rop, rop, pN); fmpz_clear(f); } } fmpz_clear(pN); }