//========================================================================
//========================= real roots  ==================================
//========================================================================

int i;
ring R=0,x,dp;
poly p=x+1;
for(i=2;i<=20;i++){p=p*(x+i);}
p=p+1/2^23*x19;
p;

LIB"solve.lib";
list L=solve(p,1000,0,1000,"nodisplay");
 def SS=L[1];
 setring SS;
 SOL;

setring R;
LIB "rootsur.lib";
nrroots(p);     


kill R;

//---------------------- A problem from economy  -------------------------
/*
ring R = 0,x(1..22),dp;
ideal I =
-1+x(1)^5*x(4)*x(13),
-1+x(2)^5*x(4)*x(14),
-1+x(3)^5*x(4)*x(15),
-1+x(5)^3*x(8)*x(13),
-1+x(6)^3*x(8)*x(14),
-1+x(7)^3*x(8)*x(15),
-1+x(9)^4*x(12)*x(13),
-1+x(10)^4*x(12)*x(14),
-1+x(11)^4*x(12)*x(15),
5+2*x(16)-x(1)*x(13)-x(2)*x(14)-x(3)*x(15),
3+5*x(16)-x(5)*x(13)-x(6)*x(14)-x(7)*x(15),
(x(1)+x(5)+x(9))^3-x(17)^2*x(18),
(x(2)+x(6)+x(10))^2-x(19)*x(20),
(x(3)+x(7)+x(11))^2-4*x(21)*x(22),
x(17)+x(19)+x(21)-10,
x(18)+x(20)+x(22)-10,
8*x(13)^3*x(18)-27*x(16)^3*x(17),
x(13)^3*x(17)^2-27*x(18)^2,
x(14)^2*x(20)-4*x(16)^2*x(19),
x(14)^2*x(19)-4*x(20),
x(15)^2*x(22)-x(16)^2*x(21),
x(15)^2*x(21)-x(22);


[1]:
   1.3052038473760073256623306244817723267588
[2]:
   1.3015615481354881445308859481238941838024
[3]:
   1.4951016092706294276032406269979249042168
[4]:
   0.1177781622069377483217973542947036297245
[5]:
   1.5952142884773910674527976400006334736637
[6]:
   1.587801855753990046379741047911189107525
[7]:
   2.000504981126595096852145097851486893749
[8]:
   0.1099004338442310860992554824336595075249
[9]:
   1.0036890295014795948077757476240053622462
[10]:
   1.0001891358877747542338111572455677741369
[11]:
   1.189432036746165160621093881016603209741
[12]:
   0.4396021328049473249197481732025006287596
[13]:
   2.2415245518985307429141711235276676775061
[14]:
   2.2730640719635375307986935544224207811234
[15]:
   1.1365320359817687653993467772112103905617
[16]:
   1.2917050688128645316413842694628547496947
[17]:
   4.5165893647991132091724493807295711813949
[18]:
   2.9170506881286453164138426946285474969468
[19]:
   3.4222990788089370518402204519976365817761
[20]:
   4.4206010670911009313877589338804922465161
[21]:
   2.061111556391949738987330167272792236829
[22]:
   2.6623482447802537521983983714909602565371
*/


//========================================================================
//==================       genus of a curve       ========================
//========================================================================
//Needs: Puiseux expansion,Primary decomposition,Field extensions
//Newton polygon. 

LIB "normal.lib";

//Rob Koelman
ring r=0,(x,y,z),dp;//genus 10  with 26 cusps
ideal I=
761328152*x^6*z^4-5431439286*x^2*y^8+2494*x^2*z^8+228715574724*x^6*y^4+
 9127158539954*x^10-15052058268*x^6*y^2*z^2+3212722859346*x^8*y^2-
 134266087241*x^8*z^2-202172841*y^8*z^2-34263110700*x^4*y^6-6697080*y^6*z^4-
 2042158*x^4*z^6-201803238*y^10+12024807786*x^4*y^4*z^2-128361096*x^4*y^2*z^4+
 506101284*x^2*z^2*y^6+47970216*x^2*z^4*y^4+660492*x^2*z^6*y^2-
 z^10-474*z^8*y^2-84366*z^6*y^4;

genus(I);

//======================================================
//          Groebner basis can be complicated
//======================================================

option(redSB);

ring r=0,(x,y),lp;
ideal i=5x4+3x2y9+y9,9x3y8+9xy8+11y10;
ideal j=std(i);
simplify(j,1);