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\fbox{\huge {\color{green}Operations on Integers}}

idiv, irem (division with remainder), ishift (bit shifting), iand, ior, ixor (logical operations), igcd,(GCD by various methods such as Euclid's algorithm and the accelerated GCD algorithm), fac (factorial), inv (inverse modulo an integer), random (random number generator by the Mersenne twister algorithm).

\fbox{\huge {\color{green}Ground Fields}}

Arithmetics on various fields: the rationals, ${\bf Q}(\alpha_1,\alpha_2,\ldots,\alpha_n)$ ($\alpha_i$ is algebraic over ${\bf Q}(\alpha_1,\ldots,\alpha_{i-1})$), $GF(p)$ ($p$ is a prime of arbitrary size), $GF(2^n)$.

\fbox{\huge {\color{green}Operations on Polynomials}}

sdiv , srem (division with remainder), ptozp (removal of the integer content), diff (differentiation), gcd (GCD over the rationals), res (resultant), subst (substitution), umul (fast multiplication of dense univariate polynomials by a hybrid method with Karatsuba and FFT+Chinese remainder), urembymul_precomp (fast dense univariate polynomial division with remainder by the fast multiplication and the precomputed inverse of a divisor),

\fbox{\huge {\color{green}Polynomial Factorization}} fctr (factorization over the rationals), fctr_ff (univariate factorization over finite fields), af (univariate factorization over algebraic number fields), sp (splitting field computation).

\fbox{\huge {\color{green} Groebner basis}}

dp_gr_main (Groebner basis computation of a polynomial ideal over the rationals by the trace lifting), dp_gr_mod_main (Groebner basis over small finite fields), tolex (Modular change of ordering for a zero-dimensional ideal), tolex_gsl (Modular rational univariate representation for a zero-dimensional ideal), dp_f4_main ($F_4$ over the rationals), dp_f4_mod_main ($F_4$ over small finite fields).

\fbox{\huge {\color{green} Ideal Decomposition}}

primedec (Prime decomposition of the radical), primadec (Primary decomposition of ideals by Shimoyama/Yokoyama algorithm).

\fbox{\huge {\color{green} Quantifier Elimination}}

qe (real quantifier elimination in a linear and quadratic first-order formula), simpl (heuristic simplification of a first-order formula).

[0] MTP2 = ex([x11,x12,x13,x21,x22,x23,x31,x32,x33],
x11+x12+x13 @== a1 @&& x21+x22+x23 @== a2 @&& x31+x32+x33 @== a3 
@&& x11+x21+x31 @== b1 @&& x12+x22+x32 @== b2 @&& x13+x23+x33 @== b3
@&& 0 @<= x11 @&& 0 @<= x12 @&& 0 @<= x13 @&& 0 @<= x21
@&& 0 @<= x22 @&& 0 @<= x23 @&& 0 @<= x31 @&& 0 @<= x32 @&& 0 @<= x33)$
[1] TSOL= a1+a2+a3@=b1+b2+b3 @&& a1@>=0 @&& a2@>=0 @&& a3@>=0
@&& b1@>=0 @&& b2@>=0 @&& b3@>=0$
[2] QE_MTP2 = qe(MTP2)$
[3] qe(all([a1,a2,a3,b1,b2,b3],QE_MTP2 @equiv TSOL));
@true

\fbox{\huge {\color{green} Visualization of curves}}

plot (plotting of a univariate function), ifplot (plotting zeros of a bivariate polynomial), conplot (contour plotting of a bivariate polynomial function).

\fbox{\huge {\color{green} Miscellaneous functions}}

det (determinant), qsort (sorting of an array by the quick sort algorithm), eval (evaluation of a formula containing transcendental functions such as sin, cos, tan, exp, log) roots (finding all roots of a univariate polynomial), lll (computation of an LLL-reduced basis of a lattice).

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Nobuki Takayama Heisei 25.9.20.