F(s) = det |
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1! 2! ...(p-1)! (mp)! d(m,p) := --------------------- m!(m+1)!....(m+p-1)!Shapiro's conjecture is the following (in this context):
m,p | # sols. | Volume of Newton Polytope | Bezout number |
2,2 | 2 | 4 | 16 |
2,3 | 5 | 17 | 64 |
2,4 | 14 | 66 | 256 |
2,5 | 42 | 247 | 1024 |
interface(quiet=true):with(linalg): K := stack(matrix([[a,b,c],[d,e,f]]),band([1],3)): Eq := s -> concat(matrix([ [1, 0],[s , 1],[s^2, 2*s],[s^3,3*s^2],[s^4,4*s^3]]),K): equations :=: Index :=1,2,3,4,5,6; # Your values here! for ii in Index do equations := equations union det(Eq(ii)): od: for ee in equations do lprint(ee,`,`);od; lprint(`0;`);
#This case has 14 roots, all real (m,p)=(2,4) with(linalg): K := stack(matrix([[a,b,c,d],[e,f,g,h]]),band([1],4)): Eq := s -> concat(transpose(matrix([ [100000, 10000*s , 1000*s^2 , 100*s^3 , 10*s^4 , s^5 ], [0 , 10000 , 2000*s , 300*s^2 , 40*s^3 , 5*s^4]])),K): equations :=: Index :=1, 2, 3, 4, 5, 6, 7, 8; #Your values here! for ii in Index do equations := equations union det(Eq(ii)): od: for ee in equations do lprint(ee,`,`);od; lprint(`0;`); od;
#This case has 42 real roots (m,p)=(2,5) interface(quiet=true):with(linalg): K := stack(matrix([[a,b,c,d,x],[e,f,g,h,y]]),band([1],5)): Eq := s -> concat(transpose(matrix([ [1000000,100000*s,10000*s^2,1000*s^3,100*s^4,10*s^5, s^6], [ 0 ,100000 , 20000*s ,3000*s^2,400*s^3,50*s^4,6*s^5]])),K): equations :=: Index :=1, 2, 3,4, 5,6, 7, 8, 9, 10; #Your values here! for ii in Index do equations := equations union det(Eq(ii)): od: for ee in equations do lprint(ee,`,`);od; lprint(`0;`);
# This case has 42 real roots (m,p)=(3,3) interface(quiet=true): with(linalg): K := stack(matrix([[a,b,c],[d,e,f],[g,h,y]]),band([1],3)): Eq := s -> concat(transpose(matrix([ [1, 1*s, 1*s^2, 1*s^3, 1*s^4, 1*s^5 ], [0, 1 , 2*s , 3*s^2, 4*s^3, 5*s^4 ], [0, 0 , 1 , 3*s , 6*s^2, 10*s^3]])),K): equations :=: Index :=1,2,3,4,5,6,7,8,9; #Your values here! for ii in Index do equations := equations union det(Eq(ii)): od: for ee in equations do lprint(ee,`,`);od; lprint(`0;`);
# Now come some systemd that I cannot solve using Grobner bases # The first has 132 roots, and the conjecture is that they are all real. # Are they? (m,p)=(2,6) interface(quiet=true):with(linalg): K := stack(matrix([[a,b,c,d,e,w],[x,y,z,f,g,h]]),band([1],6)): Eq := s -> concat(transpose(matrix([ [1, s, s^2, s^3, s^4, s^5, s^6, s^7], [0, 1, 2*s, 3*s^2, 4*s^3, 5*s^4, 6*s^5, 7*s^6]])),K): equations :=: Index :=-6,-5,-4,-3,-2,1,2,3,4,5,8,9; #Your values here! for ii in Index do equations := equations union det(Eq(ii)): od: convert(equations,list);
# This one has 462, or so roots are they all real? (m,p)=(3,4) # By the way, this is the first case where I do not know if it is # possible to find mp real p-planes in R^m+p for which the # d(m,p) m-planes which meet all of them are real) interface(quiet=true): with(linalg): K := stack(matrix([[a,b,c,d],[e,f,g,h],[x,y,z,w]]),band([1],4)): Eq := s -> concat(transpose(matrix([ [64,32*s,16*s^2, 8*s^3, 4*s^4, 2*s^5 , 1*s^6], [ 0,32 ,32*s , 24*s^2, 16*s^3, 10*s^4 , 6*s^5], [ 0, 0 ,16 , 24*s , 24*s^2, 20*s^3 , 15*s^4]])),K): equations :=: Index :=-4,-3,-2,-1,1,2,3,4,5,6,7,8; #Your values here! for ii in Index do equations := equations union det(Eq(ii)): od: for ee in equations do lprint(ee,`,`);od; lprint(`0;`);