Now we revisit the process of Grobner bases and finding solutions of a system of polynomial equations using MAPLE.restart; F:=[x^2+y^2-1,x^2-y^2-z,x+y-z^2];

NiM+SSJGRzYiNyUsKCokSSJ4R0YlIiIjIiIiKiRJInlHRiVGKkYrISIiRissKEYoRitGLEYuSSJ6R0YlRi4sKEYpRitGLUYrKiRGMEYqRi4=

with(Groebner);

NiM3OEktR0JfSW50ZXJuYWxzRzYiSSpNdWxNYXRyaXhHRiVJKVNldEJhc2lzR0YlSSpmZ2xtX2FsZ29HRiVJJ2diYXNpc0dGJUknZ3NvbHZlR0YlSStoaWxiZXJ0ZGltR0YlSSxoaWxiZXJ0cG9seUdGJUkuaGlsYmVydHNlcmllc0dGJUktaW50ZXJfcmVkdWNlR0YlSSppc19maW5pdGVHRiVJLGlzX3NvbHZhYmxlR0YlSSpsZWFkY29lZmZHRiVJKGxlYWRtb25HRiVJKWxlYWR0ZXJtR0kqcHJvdGVjdGVkR0Y0SShub3JtYWxmR0YlSS9wcmV0ZW5kX2diYXNpc0dGJUkncmVkdWNlR0YlSSZzcG9seUdGJUkqdGVybW9yZGVyR0YlSSp0ZXN0b3JkZXJHRiVJKXVuaXZwb2x5R0Yl

G:=gbasis(F,plex(z,y,x));

NiM+SSJHRzYiNyUsMCokSSJ4R0YlIiIjISIkKiRGKSIiJiEiJSokRikiIilGMCokRikiIichIzsqJEYpIiIlIiM3KiRGKSIiJEY1RikhIiIsLEYpIiIiSSJ5R0YlRjtGNEYuRihGNUY5RjssKEkiekdGJUY7RighIiNGO0Y7

factor(G);

NiM3JSoqSSJ4RzYiIiIiLCZGJUYnISIiRidGJywmKiRGJSIiI0YsRilGJ0YnLCwqJEYlIiIlRi8qJEYlIiIkRi9GKyEiI0YlISIlRilGJ0YnLCxGJUYnSSJ5R0YmRidGLkYzRitGL0YpRicsKEkiekdGJkYnRitGMkYnRic=

_EnvExplicit:=true; solve(4*x^4+4*x^3-2*x^2-4*x-1,x);

NiM+SS1fRW52RXhwbGljaXRHNiJJJXRydWVHSSpwcm90ZWN0ZWRHRic=NiYsKCMhIiIiIiUiIiIqJCIiJiNGJyIiIyNGJ0YmKiQsJkYrRidGKEYrRipGLCwoRiRGJ0YoRixGLUYkLChGJEYnRihGJCokLCZGK0YnRighIiNGKkYsLChGJEYnRihGJEYxRiQ=

This environment variable will try to express with radicals all those roots for which is possible (e.g. in polynomials of degree less than or equal to four). When the variable is set to false then roots are handled symbolically.assign({solve(G[3],{z}), solve(G[2],{y})});rootlist:=[solve(G[1])];

NiM+SSlyb290bGlzdEc2IjcqIiIhIiIiLCQqJCIiIyNGKEYrRiwsJEYqIyEiIkYrLCgjRi8iIiVGKCokIiImRiwjRihGMiokLCZGK0YoRjNGK0YsRjUsKEYxRihGM0Y1RjZGMSwoRjFGKEYzRjEqJCwmRitGKEYzISIjRixGNSwoRjFGKEYzRjFGOkYx

for x in rootlist do whatever:=simplify(['x'=x,'y'=y, 'z'=z])): if simplify(subs(op(whatever),F))=[0,0,0] then print("valid solution"); end if;od:

Error, `)` unexpected

We can also (A) decide whether a system has solutions(B) decide whether there are finitely many or infinitely many solutions(C) decide exactly how many solutions are there.LEMMA: (Hilbert's Nullstellensatz) A system of polynomial equations has complex solution if and only if a Grobner basis for the ideal generated by the polynomials is not equal [1].restart; with(Groebner); gbasis([x+x*y^2-1,x^2*y+y-1,x^2+y^2-1],tdeg(x,y));

NiM3OEktR0JfSW50ZXJuYWxzRzYiSSpNdWxNYXRyaXhHRiVJKVNldEJhc2lzR0YlSSpmZ2xtX2FsZ29HRiVJJ2diYXNpc0dGJUknZ3NvbHZlR0YlSStoaWxiZXJ0ZGltR0YlSSxoaWxiZXJ0cG9seUdGJUkuaGlsYmVydHNlcmllc0dGJUktaW50ZXJfcmVkdWNlR0YlSSppc19maW5pdGVHRiVJLGlzX3NvbHZhYmxlR0YlSSpsZWFkY29lZmZHRiVJKGxlYWRtb25HRiVJKWxlYWR0ZXJtR0kqcHJvdGVjdGVkR0Y0SShub3JtYWxmR0YlSS9wcmV0ZW5kX2diYXNpc0dGJUkncmVkdWNlR0YlSSZzcG9seUdGJUkqdGVybW9yZGVyR0YlSSp0ZXN0b3JkZXJHRiVJKXVuaXZwb2x5R0YlNiM3IyIiIg==

LEMMA: A system of equations has a finite number of solutions if and only if anyGrobner basis of the polynomials has the property that for every variablex_i there exists a polynomial such that its leading term with respect to thechosen term is a power of x_i.polys:=[c*x+x*y^2+x*z^2-1,c*y+y*x^2+y*z^2-1,c*z+z*x^2+z*y^2-1];

NiM+SSZwb2x5c0c2IjclLCoqJkkiY0dGJSIiIkkieEdGJUYqRioqJkYrRipJInlHRiUiIiNGKiomRitGKkkiekdGJUYuRiohIiJGKiwqKiZGKUYqRi1GKkYqKiZGK0YuRi1GKkYqKiZGLUYqRjBGLkYqRjFGKiwqKiZGKUYqRjBGKkYqKiZGMEYqRitGLkYqKiZGMEYqRi1GLkYqRjFGKg==

with(Groebner): is_finite(gbasis(polys,tdeg(x,y,z))); sys:=gsolve(polys,[c,x,y,z]);

NiNJJmZhbHNlR0kqcHJvdGVjdGVkR0YkNiM+SSRzeXNHNiI8JTclNyUsJkkieUdGJSIiIkkiekdGJSEiIiwoKiZJInhHRiVGK0YsIiIjRisqJkYsRitGMEYxRitGLUYrLChJImNHRiVGKyomRixGK0YwRitGLSokRixGMUYrLUklcGxleEdGJTYmRjRGMEYqRiw8IywmRjBGLUYsRis3JTclRiksJkYwRitGLEYtLCgqJkY0RitGLEYrRisqJEYsIiIkRjFGLUYrRjc8IjclNyUsKComRipGK0YsRjFGKyomRixGK0YqRjFGK0YtRissLComRjBGK0YqRjFGLUYvRi1GK0YrKiRGMEZCRisqKEYsRitGKkYrRjBGK0YtLChGNEYrKiRGMEYxRisqJkYqRitGLEYrRi1GNzwjLCZGLEYrRipGLQ==

for s in sys do solve({op(s[1])},{x,y,z}) end do;

NiM8JS9JInhHNiIsJComLCghIiIiIiIqJkkiY0dGJkYrLUknUm9vdE9mRzYkSSpwcm90ZWN0ZWRHRjFJKF9zeXNsaWJHRiY2IywqKiZGLUYrSSNfWkdGJiIiIyIiJCokRjYiIiVGNyokRi1GN0YrRjZGKkYrRjcqJEYuRjhGN0YrRi1GKkYqL0kieUdGJkYuL0kiekdGJkYuNiM8JS9JInpHNiItSSdSb290T2ZHNiRJKnByb3RlY3RlZEdGKkkoX3N5c2xpYkdGJjYjLCgqJkkiY0dGJiIiIkkjX1pHRiZGMEYwKiRGMSIiJCIiIyEiIkYwL0kieUdGJkYnL0kieEdGJkYn

NiQ8JS9JInpHNiItSSdSb290T2ZHNiRJKnByb3RlY3RlZEdGKkkoX3N5c2xpYkdGJjYjLCoqJEkjX1pHRiYiIiMiIiJJImNHRiZGMSomLUYoNiMsKComRjJGMUYvRjFGMUYxRjEqJEYvIiIkRjFGMUYvRjFGMSokRjRGMEYxL0kieUdGJiomLCZGMkYxRjpGMUYxRichIiIvSSJ4R0YmRjQ8JS9GPComLCZGMkYxKiQtRig2IywqKiZGMkYxRi9GMEY5KiRGLyIiJUYwKiRGMkYwRjFGL0Y/RjBGMUYxLUYoNiMsKkZKRjEqJiwoKiRGR0Y5RjAqJkYyRjFGR0YxRjFGP0YxRjFGL0YxRjEqJkYyRjFGR0YwRjFGTUYxRj8vRkFGRy9GJUZO

G:=gbasis(polys,plex(c,x,y,z));

NiM+SSJHRzYiNygsKiomSSJ5R0YlIiIiSSJ6R0YlIiIkISIiKiZGK0YqRilGLEYqRitGKkYpRi0sKiomRitGKkkieEdGJUYsRipGMUYtKiZGK0YsRjFGKkYtRitGKiwqKiZGKUYqRjFGLEYqRjFGLSomRilGLEYxRipGLUYpRiosKiomSSJjR0YlRipGK0YqRioqJkYrRipGMSIiI0YqKiZGK0YqRilGOkYqRi1GKiwqKiZGOEYqRilGKkYqKiZGMUY6RilGKkYqKiZGKUYqRitGOkYqRi1GKiwqKiZGOEYqRjFGKkYqKiZGMUYqRilGOkYqKiZGMUYqRitGOkYqRi1GKg==

is_finite(G);

NiNJJmZhbHNlR0kqcHJvdGVjdGVkR0Yk

What MAPLE uses to determine whether the number of solutions is finiteis the following lemma:LEMMA: For a system of polynomials the number of solutions is finite ifand only if the set of monomials which are not multiples of a the leadingmonomials of a Grobner basis is finite.Now, if the number of solutions is infinite there is a nice way we can knowthe dimension of the solution space (i.e. how man free parameters shouldthere be?).hilbertdim(polys,tdeg(c,x,y,z));

NiMiIiI=

Suppose now we want to actually count the number of solutions:LEMMA: Suppose that the polynomials f_1, f_2, ..., f_s have a finite number of solutions. Then, the number of solutions (counted with multiplicities and solution at infinity) is equal to the cardinality of the sets of monomials that are no multiples of leading monomials of any Grobner basis (any term order).G:=gbasis(polys, tdeg(x,y,z));

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

LM:=map(f -> leadmon(f,tdeg(x,y,z))[2],G);

NiM+SSNMTUc2IjctKiZJInpHRiUiIiJJInhHRiUiIiMqJkYqRilJInlHRiVGKyomRipGK0YtRikqJkYtRitGKEYrKiZGKEYpRi0iIiQqJEYtIiIlKiRGKkYzKiRGKCIiJiomRi1GKUYoRjMqJkYqRilGKEYzKihGLUYpRipGKUYoRjE=

S:=expand(series(1/((1-x*u)*(1-y*u)*(1-z*u)),u,5));

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

L:=[op(subs(u=1, convert(S,polynom)))]; nops(L);

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with(Groebner); notdiv:={}:

NiM3OEktR0JfSW50ZXJuYWxzRzYiSSpNdWxNYXRyaXhHRiVJKVNldEJhc2lzR0YlSSpmZ2xtX2FsZ29HRiVJJ2diYXNpc0dGJUknZ3NvbHZlR0YlSStoaWxiZXJ0ZGltR0YlSSxoaWxiZXJ0cG9seUdGJUkuaGlsYmVydHNlcmllc0dGJUktaW50ZXJfcmVkdWNlR0YlSSppc19maW5pdGVHRiVJLGlzX3NvbHZhYmxlR0YlSSpsZWFkY29lZmZHRiVJKGxlYWRtb25HRiVJKWxlYWR0ZXJtR0kqcHJvdGVjdGVkR0Y0SShub3JtYWxmR0YlSS9wcmV0ZW5kX2diYXNpc0dGJUkncmVkdWNlR0YlSSZzcG9seUdGJUkqdGVybW9yZGVyR0YlSSp0ZXN0b3JkZXJHRiVJKXVuaXZwb2x5R0Yl

for m in L do if normalf(m,LM,plex(x,y,z))<>0 then notdiv:=notdiv union {m}: fi:od; print("number of solutions",nops(notdiv));

NiRRNG51bWJlcn5vZn5zb2x1dGlvbnM2IiIjQA==

with(Ore_algebra): A:=poly_algebra(x,y,z);

NiM+SSJBRzYiSSxPcmVfYWxnZWJyYUdGJQ==

with(Groebner): T:=termorder(A,plex(x,y,z)):sort(L,(t1,t2)->testorder(t1,t2,T));

NiM3RSIiIkkiekc2IiokRiUiIiMqJEYlIiIkKiRGJSIiJUkieUdGJiomRi1GJEYlRiQqJkYtRiRGJUYoKiZGLUYkRiVGKiokRi1GKComRiVGJEYtRigqJkYtRihGJUYoKiRGLUYqKiZGJUYkRi1GKiokRi1GLEkieEdGJiomRiVGJEY3RiQqJkY3RiRGJUYoKiZGJUYqRjdGJComRi1GJEY3RiQqKEYlRiRGLUYkRjdGJCooRi1GJEY3RiRGJUYoKiZGN0YkRi1GKCooRjdGJEYlRiRGLUYoKiZGLUYqRjdGJCokRjdGKComRiVGJEY3RigqJkY3RihGJUYoKiZGN0YoRi1GJCooRjdGKEYtRiRGJUYkKiZGN0YoRi1GKCokRjdGKiomRiVGJEY3RioqJkYtRiRGN0YqKiRGN0Ys

T:=termorder(A,tdeg(x,y,z));

NiM+SSJURzYiSSt0ZXJtX29yZGVyR0Yl

sort(L,(t1,t2)->testorder(t1,t2,T));

NiM3RSIiIkkiekc2IkkieUdGJkkieEdGJiokRiUiIiMqJkYnRiRGJUYkKiZGJUYkRihGJCokRidGKiomRidGJEYoRiQqJEYoRioqJEYlIiIkKiZGJ0YkRiVGKiomRihGJEYlRioqJkYlRiRGJ0YqKihGJUYkRidGJEYoRiQqJkYlRiRGKEYqKiRGJ0YxKiZGKEYkRidGKiomRihGKkYnRiQqJEYoRjEqJEYlIiIlKiZGJ0YkRiVGMSomRiVGMUYoRiQqJkYnRipGJUYqKihGJ0YkRihGJEYlRioqJkYoRipGJUYqKiZGJUYkRidGMSooRihGJEYlRiRGJ0YqKihGKEYqRidGJEYlRiQqJkYlRiRGKEYxKiRGJ0Y8KiZGJ0YxRihGJComRihGKkYnRioqJkYnRiRGKEYxKiRGKEY8

T:=termorder(A,wdeg([3,2,1],[x,y,z])):sort(L,(t1,t2)->testorder(t1,t2,T));

NiM3RSIiIkkiekc2IiokRiUiIiNJInlHRiYqJEYlIiIkKiZGKUYkRiVGJEkieEdGJiokRiUiIiUqJkYpRiRGJUYoKiZGJUYkRi1GJCokRilGKComRilGJEYlRisqJkYtRiRGJUYoKiZGJUYkRilGKComRilGJEYtRiQqJkYlRitGLUYkKiZGKUYoRiVGKCooRiVGJEYpRiRGLUYkKiRGKUYrKiRGLUYoKihGKUYkRi1GJEYlRigqJkYlRiRGKUYrKiZGJUYkRi1GKComRi1GJEYpRigqJkYtRihGJUYoKihGLUYkRiVGJEYpRigqJEYpRi8qJkYtRihGKUYkKihGLUYoRilGJEYlRiQqJkYpRitGLUYkKiRGLUYrKiZGJUYkRi1GKyomRi1GKEYpRigqJkYpRiRGLUYrKiRGLUYv

with(linalg): with(Groebner);

NiM3OEktR0JfSW50ZXJuYWxzRzYiSSpNdWxNYXRyaXhHRiVJKVNldEJhc2lzR0YlSSpmZ2xtX2FsZ29HRiVJJ2diYXNpc0dGJUknZ3NvbHZlR0YlSStoaWxiZXJ0ZGltR0YlSSxoaWxiZXJ0cG9seUdGJUkuaGlsYmVydHNlcmllc0dGJUktaW50ZXJfcmVkdWNlR0YlSSppc19maW5pdGVHRiVJLGlzX3NvbHZhYmxlR0YlSSpsZWFkY29lZmZHRiVJKGxlYWRtb25HRiVJKWxlYWR0ZXJtR0kqcHJvdGVjdGVkR0Y0SShub3JtYWxmR0YlSS9wcmV0ZW5kX2diYXNpc0dGJUkncmVkdWNlR0YlSSZzcG9seUdGJUkqdGVybW9yZGVyR0YlSSp0ZXN0b3JkZXJHRiVJKXVuaXZwb2x5R0Yl

T:=termorder(A,'matrix'([[1,1,1],[1,0,0],[0,1,0]],[x,y,z])):Lgrlex2:=sort(L, (t1,t2) ->testorder(t1,t2,T));

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