{
"cells": [
{
"cell_type": "markdown",
"source": [
"# Degree gap versus staircase gap"
],
"metadata": {}
},
{
"outputs": [
{
"output_type": "execute_result",
"data": {
"text/plain": "Random.TaskLocalRNG()"
},
"metadata": {},
"execution_count": 1
}
],
"cell_type": "code",
"source": [
"using TypedPolynomials\n",
"using SemialgebraicSets\n",
"function system(k, solver)\n",
" @polyvar x y\n",
" return @set x == y^k && y^k == 0 solver\n",
"end\n",
"using Random\n",
"Random.seed!(80)"
],
"metadata": {},
"execution_count": 1
},
{
"cell_type": "markdown",
"source": [
"Let's start with the classical approach with all columns up to degree 5 and waiting for the gap:"
],
"metadata": {}
},
{
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"[ Info: Added 2 rows to complete columns up to degree 4\n",
"[ Info: Nullspace of dimensions (15, 13) computed from Macaulay matrix of dimension (2, 15) in 6.448e-5 seconds.\n",
"[ Info: Added 4 rows to complete columns up to degree 5\n",
"[ Info: Nullspace of dimensions (21, 15) computed from Macaulay matrix of dimension (6, 21) in 4.2128e-5 seconds.\n",
"[ Info: Added 6 rows to complete columns up to degree 6\n",
"[ Info: Nullspace of dimensions (28, 16) computed from Macaulay matrix of dimension (12, 28) in 5.0625e-5 seconds.\n",
"[ Info: Added 8 rows to complete columns up to degree 7\n",
"[ Info: Nullspace of dimensions (36, 16) computed from Macaulay matrix of dimension (20, 36) in 6.1073e-5 seconds.\n",
"[ Info: Found multiplication matrices with commutation error of `0.5555434329742673` which is larger than the tolerance of `0.0001220703125`. Adding this to the equations and continuing.\n",
"[ Info: Found multiplication matrices with commutation error of `0.14137366352981615` which is larger than the tolerance of `0.0001220703125`. Adding this to the equations and continuing.\n",
"[ Info: Found multiplication matrices with commutation error of `0.32931221645105807` which is larger than the tolerance of `0.0001220703125`. Adding this to the equations and continuing.\n",
"[ Info: Found 1 real solution\n"
]
}
],
"cell_type": "code",
"source": [
"using MacaulayMatrix\n",
"sys_gap = system(4, MacaulayMatrix.Solver(sparse_columns = false, wait_for_gap = true))\n",
"solutions = collect(sys_gap)\n",
"nothing #hide"
],
"metadata": {},
"execution_count": 2
},
{
"cell_type": "markdown",
"source": [
"We find the expected solution"
],
"metadata": {}
},
{
"outputs": [
{
"output_type": "execute_result",
"data": {
"text/plain": "1-element Vector{Vector{Float64}}:\n [-1.7687461110662856e-16, -0.39155425716877207]"
},
"metadata": {},
"execution_count": 3
}
],
"cell_type": "code",
"source": [
"solutions"
],
"metadata": {},
"execution_count": 3
},
{
"cell_type": "markdown",
"source": [
"If we don't wait for the gap, we get it earlier:"
],
"metadata": {}
},
{
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"[ Info: Added 2 rows to complete columns up to degree 4\n",
"[ Info: Nullspace of dimensions (15, 13) computed from Macaulay matrix of dimension (2, 15) in 3.4384e-5 seconds.\n",
"┌ Warning: `LAPACK.trsen!` throwed an exception for `[3, 4]` so using default tolerance `2.220446049250313e-16`\n",
"└ @ SemialgebraicSets ~/.julia/packages/SemialgebraicSets/BoKut/src/schur.jl:22\n",
"[ Info: Found 1 real solution\n"
]
}
],
"cell_type": "code",
"source": [
"using MacaulayMatrix\n",
"sys = system(4, MacaulayMatrix.Solver(sparse_columns = false))\n",
"solutions = collect(sys)\n",
"nothing #hide"
],
"metadata": {},
"execution_count": 4
},
{
"cell_type": "markdown",
"source": [
"We find the expected solution"
],
"metadata": {}
},
{
"outputs": [
{
"output_type": "execute_result",
"data": {
"text/plain": "1-element Vector{Vector{Float64}}:\n [-1.962615573354718e-16, -1.1927117914146596e-16]"
},
"metadata": {},
"execution_count": 5
}
],
"cell_type": "code",
"source": [
"solutions"
],
"metadata": {},
"execution_count": 5
},
{
"cell_type": "markdown",
"source": [
"But we can actually also just use the 2 columns that are actually used:"
],
"metadata": {}
},
{
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"[ Info: Added 2 rows to complete columns up to degree 4\n",
"[ Info: Nullspace of dimensions (2, 0) computed from Macaulay matrix of dimension (2, 2) in 1.7833e-5 seconds.\n",
"[ Info: Found 1 real solution\n"
]
}
],
"cell_type": "code",
"source": [
"sys_sparse = system(4, MacaulayMatrix.Solver())\n",
"solutions = collect(sys_sparse)\n",
"nothing #hide"
],
"metadata": {},
"execution_count": 6
},
{
"cell_type": "markdown",
"source": [
"We find the expected solution"
],
"metadata": {}
},
{
"outputs": [
{
"output_type": "execute_result",
"data": {
"text/plain": "1-element Vector{Vector{Float64}}:\n [0.0, 0.0]"
},
"metadata": {},
"execution_count": 7
}
],
"cell_type": "code",
"source": [
"solutions"
],
"metadata": {},
"execution_count": 7
},
{
"cell_type": "markdown",
"source": [
"## In more details"
],
"metadata": {}
},
{
"outputs": [
{
"output_type": "execute_result",
"data": {
"text/plain": "MacaulayMatrix matrix solver. Last iteration considered:\n0×0 Macaulay matrix for polynomials:\n x - y^4\n y^4\nThe row shifts are:\nMonomialBasis([])\nThe column basis is:\nMonomialBasis([])\nCurrent status is OPTIMIZE_NOT_CALLED\nHistory of iterations:\n\u001b[1m0×3 DataFrame\n\u001b[1m Row │\u001b[1m nullity \u001b[1m num_rows \u001b[1m num_cols\n │\u001b[90m Int64 \u001b[90m Int64 \u001b[90m Int64\n─────┴─────────────────────────────"
},
"metadata": {},
"execution_count": 8
}
],
"cell_type": "code",
"source": [
"solver_gap = init(sys_gap, sys_gap.solver)"
],
"metadata": {},
"execution_count": 8
},
{
"cell_type": "markdown",
"source": [
"After one step, we don't have the solution yet"
],
"metadata": {}
},
{
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"[ Info: Added 2 rows to complete columns up to degree 4\n",
"[ Info: Nullspace of dimensions (15, 13) computed from Macaulay matrix of dimension (2, 15) in 4.8229e-5 seconds.\n"
]
}
],
"cell_type": "code",
"source": [
"step!(solver_gap)"
],
"metadata": {},
"execution_count": 9
},
{
"cell_type": "markdown",
"source": [
"After the second step, still no solution"
],
"metadata": {}
},
{
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"[ Info: Added 4 rows to complete columns up to degree 5\n",
"[ Info: Nullspace of dimensions (21, 15) computed from Macaulay matrix of dimension (6, 21) in 3.0326e-5 seconds.\n"
]
}
],
"cell_type": "code",
"source": [
"step!(solver_gap)"
],
"metadata": {},
"execution_count": 10
},
{
"cell_type": "markdown",
"source": [
"After the third step, still no solution"
],
"metadata": {}
},
{
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"[ Info: Added 6 rows to complete columns up to degree 6\n",
"[ Info: Nullspace of dimensions (28, 16) computed from Macaulay matrix of dimension (12, 28) in 4.1838e-5 seconds.\n"
]
}
],
"cell_type": "code",
"source": [
"step!(solver_gap)"
],
"metadata": {},
"execution_count": 11
},
{
"cell_type": "markdown",
"source": [
"After the fourth step, we find the solution"
],
"metadata": {}
},
{
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"[ Info: Added 8 rows to complete columns up to degree 7\n",
"[ Info: Nullspace of dimensions (36, 16) computed from Macaulay matrix of dimension (20, 36) in 6.3197e-5 seconds.\n",
"[ Info: Found multiplication matrices with commutation error of `0.5555434329742673` which is larger than the tolerance of `0.0001220703125`. Adding this to the equations and continuing.\n",
"[ Info: Found multiplication matrices with commutation error of `0.14137366352981615` which is larger than the tolerance of `0.0001220703125`. Adding this to the equations and continuing.\n",
"[ Info: Found multiplication matrices with commutation error of `0.32931221645105807` which is larger than the tolerance of `0.0001220703125`. Adding this to the equations and continuing.\n",
"[ Info: Found 1 real solution\n"
]
}
],
"cell_type": "code",
"source": [
"step!(solver_gap)"
],
"metadata": {},
"execution_count": 12
},
{
"cell_type": "markdown",
"source": [
"We can inspect the solver at follows"
],
"metadata": {}
},
{
"outputs": [
{
"output_type": "execute_result",
"data": {
"text/plain": "MacaulayMatrix matrix solver. Last iteration considered:\n20×36 Macaulay matrix for polynomials:\n x - y^4\n y^4\nThe row shifts are:\nMonomialBasis([1, y, x, y^2, x*y, x^2, y^3, x*y^2, x^2*y, x^3])\nThe column basis is:\nMonomialBasis([1, y, x, y^2, x*y, x^2, y^3, x*y^2, x^2*y, x^3, y^4, x*y^3, x^2*y^2, x^3*y, x^4, y^5, x*y^4, x^2*y^3, x^3*y^2, x^4*y, x^5, y^6, x*y^5, x^2*y^4, x^3*y^3, x^4*y^2, x^5*y, x^6, y^7, x*y^6, x^2*y^5, x^3*y^4, x^4*y^3, x^5*y^2, x^6*y, x^7])\nBorderBasis with independent rows and dependent columns in:\nBasisDependence for bases:\n Standard:\n MonomialBasis([1, y, y^2, y^3])\n Corners:\n MonomialBasis([x, y^4])\n Dependent Border:\n MonomialBasis([x*y, x*y^2, x*y^3])\n\nAnd entries in a 4×5 adjoint(::Matrix{Float64}) with eltype Float64:\n -2.172825010123334e-16 1.8952692539670457e-16 … -2.7755575615628892e-17\n -1.062601985498177e-16 1.8952692539670447e-16 2.7755575615628944e-17\n 5.55111512312578e-17 -1.1102230246251565e-16 -2.775557561562892e-17\n 6.943015803029501e-34 9.720222124241336e-33 -3.471507901514769e-34\nCurrent status is OPTIMAL\nFound 1 solutions:\n [-1.7687461110662856e-16, -0.39155425716877207]\nHistory of iterations:\n\u001b[1m4×3 DataFrame\n\u001b[1m Row │\u001b[1m nullity \u001b[1m num_rows \u001b[1m num_cols\n │\u001b[90m Int64 \u001b[90m Int64 \u001b[90m Int64\n─────┼─────────────────────────────\n 1 │ 13 2 15\n 2 │ 15 6 21\n 3 │ 16 12 28\n 4 │ 16 20 36"
},
"metadata": {},
"execution_count": 13
}
],
"cell_type": "code",
"source": [
"solver_gap"
],
"metadata": {},
"execution_count": 13
},
{
"cell_type": "markdown",
"source": [
"The border dependence can be plotted for more visual inspection:"
],
"metadata": {}
},
{
"outputs": [
{
"output_type": "execute_result",
"data": {
"text/plain": "Plot{Plots.GRBackend() n=3}",
"image/png": 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]
},
"metadata": {},
"execution_count": 14
}
],
"cell_type": "code",
"source": [
"using Plots\n",
"plot(saturated_dependence(solver_gap))"
],
"metadata": {},
"execution_count": 14
},
{
"cell_type": "markdown",
"source": [
"We can see that the monomials of larger degree have not been used here.\n",
"We can see them used with `AnyDependence` instead of `StaircaseDependence`"
],
"metadata": {}
},
{
"outputs": [
{
"output_type": "execute_result",
"data": {
"text/plain": "MacaulayMatrix matrix solver. Last iteration considered:\n0×0 Macaulay matrix for polynomials:\n x - y^4\n y^4\nThe row shifts are:\nMonomialBasis([])\nThe column basis is:\nMonomialBasis([])\nCurrent status is OPTIMIZE_NOT_CALLED\nHistory of iterations:\n\u001b[1m0×3 DataFrame\n\u001b[1m Row │\u001b[1m nullity \u001b[1m num_rows \u001b[1m num_cols\n │\u001b[90m Int64 \u001b[90m Int64 \u001b[90m Int64\n─────┴─────────────────────────────"
},
"metadata": {},
"execution_count": 15
}
],
"cell_type": "code",
"source": [
"solver_gap = init(sys_gap, MacaulayMatrix.Solver(dependence = MacaulayMatrix.MM.LinearDependence, sparse_columns = false, wait_for_gap = true))"
],
"metadata": {},
"execution_count": 15
},
{
"cell_type": "markdown",
"source": [
"Let's solve again but not all the step at once:"
],
"metadata": {}
},
{
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"[ Info: Added 2 rows to complete columns up to degree 4\n",
"[ Info: Nullspace of dimensions (15, 13) computed from Macaulay matrix of dimension (2, 15) in 4.841e-5 seconds.\n",
"[ Info: Added 4 rows to complete columns up to degree 5\n",
"[ Info: Nullspace of dimensions (21, 15) computed from Macaulay matrix of dimension (6, 21) in 2.6159e-5 seconds.\n",
"[ Info: Added 6 rows to complete columns up to degree 6\n",
"[ Info: Nullspace of dimensions (28, 16) computed from Macaulay matrix of dimension (12, 28) in 3.5906e-5 seconds.\n",
"[ Info: Added 8 rows to complete columns up to degree 7\n",
"[ Info: Nullspace of dimensions (36, 16) computed from Macaulay matrix of dimension (20, 36) in 5.7116e-5 seconds.\n",
"[ Info: Found multiplication matrices with commutation error of `0.5555434329742674` which is larger than the tolerance of `0.0001220703125`. Adding this to the equations and continuing.\n",
"[ Info: Found multiplication matrices with commutation error of `0.1413736635298159` which is larger than the tolerance of `0.0001220703125`. Adding this to the equations and continuing.\n",
"[ Info: Found multiplication matrices with commutation error of `0.32931221645105857` which is larger than the tolerance of `0.0001220703125`. Adding this to the equations and continuing.\n",
"[ Info: Found 1 real solution\n"
]
},
{
"output_type": "execute_result",
"data": {
"text/plain": "1-element Vector{Vector{Float64}}:\n [-1.7687461110662856e-16, -0.3915542571687721]"
},
"metadata": {},
"execution_count": 16
}
],
"cell_type": "code",
"source": [
"solve!(solver_gap)"
],
"metadata": {},
"execution_count": 16
},
{
"cell_type": "markdown",
"source": [
"This time, we see the monomials at infinity as the blue balls outside the staircase.\n",
"We can see that the degree 4 contains no independent as it is the gap zone."
],
"metadata": {}
},
{
"outputs": [
{
"output_type": "execute_result",
"data": {
"text/plain": "Plot{Plots.GRBackend() n=4}",
"image/png": 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]
},
"metadata": {},
"execution_count": 17
}
],
"cell_type": "code",
"source": [
"using Plots\n",
"plot(saturated_dependence(solver_gap))"
],
"metadata": {},
"execution_count": 17
},
{
"cell_type": "markdown",
"source": [
"## Not waiting for the gap in more details"
],
"metadata": {}
},
{
"outputs": [
{
"output_type": "execute_result",
"data": {
"text/plain": "MacaulayMatrix matrix solver. Last iteration considered:\n0×0 Macaulay matrix for polynomials:\n x - y^4\n y^4\nThe row shifts are:\nMonomialBasis([])\nThe column basis is:\nMonomialBasis([])\nCurrent status is OPTIMIZE_NOT_CALLED\nHistory of iterations:\n\u001b[1m0×3 DataFrame\n\u001b[1m Row │\u001b[1m nullity \u001b[1m num_rows \u001b[1m num_cols\n │\u001b[90m Int64 \u001b[90m Int64 \u001b[90m Int64\n─────┴─────────────────────────────"
},
"metadata": {},
"execution_count": 18
}
],
"cell_type": "code",
"source": [
"solver = init(sys, sys.solver)"
],
"metadata": {},
"execution_count": 18
},
{
"cell_type": "markdown",
"source": [
"After one step, we find the solution"
],
"metadata": {}
},
{
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"[ Info: Added 2 rows to complete columns up to degree 4\n",
"[ Info: Nullspace of dimensions (15, 13) computed from Macaulay matrix of dimension (2, 15) in 4.8971e-5 seconds.\n",
"[ Info: Found 1 real solution\n"
]
}
],
"cell_type": "code",
"source": [
"step!(solver)"
],
"metadata": {},
"execution_count": 19
},
{
"cell_type": "markdown",
"source": [
"We can inspect `m` for more details"
],
"metadata": {}
},
{
"outputs": [
{
"output_type": "execute_result",
"data": {
"text/plain": "MacaulayMatrix matrix solver. Last iteration considered:\n2×15 Macaulay matrix for polynomials:\n x - y^4\n y^4\nThe row shifts are:\nMonomialBasis([1])\nThe column basis is:\nMonomialBasis([1, y, x, y^2, x*y, x^2, y^3, x*y^2, x^2*y, x^3, y^4, x*y^3, x^2*y^2, x^3*y, x^4])\nBorderBasis with independent rows and dependent columns in:\nBasisDependence for bases:\n Standard:\n MonomialBasis([1, y, y^2, y^3])\n Corners:\n MonomialBasis([x, y^4])\n Independent Border:\n MonomialBasis([x*y, x*y^2, x*y^3])\n\nAnd entries in a 4×2 adjoint(::Matrix{Float64}) with eltype Float64:\n -1.9626155733547191e-16 -3.925231146709439e-17\n -1.1775693440128314e-16 -1.1775693440128314e-16\n 0.0 0.0\n 0.0 0.0\nCurrent status is OPTIMAL\nFound 1 solutions:\n [-1.962615573354719e-16, 7.031051088568496e-17]\nHistory of iterations:\n\u001b[1m1×3 DataFrame\n\u001b[1m Row │\u001b[1m nullity \u001b[1m num_rows \u001b[1m num_cols\n │\u001b[90m Int64 \u001b[90m Int64 \u001b[90m Int64\n─────┼─────────────────────────────\n 1 │ 13 2 15"
},
"metadata": {},
"execution_count": 20
}
],
"cell_type": "code",
"source": [
"solver"
],
"metadata": {},
"execution_count": 20
},
{
"cell_type": "markdown",
"source": [
"The border dependence can be plotted for more visual inspection:"
],
"metadata": {}
},
{
"outputs": [
{
"output_type": "execute_result",
"data": {
"text/plain": "Plot{Plots.GRBackend() n=3}",
"image/png": 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]
},
"metadata": {},
"execution_count": 21
}
],
"cell_type": "code",
"source": [
"plot(saturated_dependence(solver))"
],
"metadata": {},
"execution_count": 21
},
{
"cell_type": "markdown",
"source": [
"Even if there is no gap, the border is complete so we can get the multiplication matrices."
],
"metadata": {}
},
{
"cell_type": "markdown",
"source": [
"## Sparse columns in more details"
],
"metadata": {}
},
{
"outputs": [
{
"output_type": "execute_result",
"data": {
"text/plain": "MacaulayMatrix matrix solver. Last iteration considered:\n0×0 Macaulay matrix for polynomials:\n x - y^4\n y^4\nThe row shifts are:\nMonomialBasis([])\nThe column basis is:\nMonomialBasis([])\nCurrent status is OPTIMIZE_NOT_CALLED\nHistory of iterations:\n\u001b[1m0×3 DataFrame\n\u001b[1m Row │\u001b[1m nullity \u001b[1m num_rows \u001b[1m num_cols\n │\u001b[90m Int64 \u001b[90m Int64 \u001b[90m Int64\n─────┴─────────────────────────────"
},
"metadata": {},
"execution_count": 22
}
],
"cell_type": "code",
"source": [
"sparse_solver = init(sys_sparse, sys_sparse.solver)"
],
"metadata": {},
"execution_count": 22
},
{
"cell_type": "markdown",
"source": [
"After one step, we find the solution"
],
"metadata": {}
},
{
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"[ Info: Added 2 rows to complete columns up to degree 4\n",
"[ Info: Nullspace of dimensions (2, 0) computed from Macaulay matrix of dimension (2, 2) in 3.0487e-5 seconds.\n",
"[ Info: Found 1 real solution\n"
]
}
],
"cell_type": "code",
"source": [
"step!(sparse_solver)"
],
"metadata": {},
"execution_count": 23
},
{
"cell_type": "markdown",
"source": [
"We can inspect `m` for more details"
],
"metadata": {}
},
{
"outputs": [
{
"output_type": "execute_result",
"data": {
"text/plain": "MacaulayMatrix matrix solver. Last iteration considered:\n2×2 Macaulay matrix for polynomials:\n x - y^4\n y^4\nThe row shifts are:\nMonomialBasis([1])\nThe column basis is:\nMonomialBasis([x, y^4])\nBorderBasis with independent rows and dependent columns in:\nBasisDependence for bases:\n Trivial Standard:\n MonomialBasis([1, y, y^2, y^3])\n Corners:\n MonomialBasis([x, y^4])\n Trivial Independent Border:\n MonomialBasis([x*y, x*y^2, x*y^3])\n\nAnd entries in a 0×2 adjoint(::Matrix{Float64}) with eltype Float64\nCurrent status is OPTIMAL\nFound 1 solutions:\n [0.0, 0.0]\nHistory of iterations:\n\u001b[1m1×3 DataFrame\n\u001b[1m Row │\u001b[1m nullity \u001b[1m num_rows \u001b[1m num_cols\n │\u001b[90m Int64 \u001b[90m Int64 \u001b[90m Int64\n─────┼─────────────────────────────\n 1 │ 0 2 2"
},
"metadata": {},
"execution_count": 24
}
],
"cell_type": "code",
"source": [
"sparse_solver"
],
"metadata": {},
"execution_count": 24
},
{
"cell_type": "markdown",
"source": [
"The border dependence can be plotted for more visual inspection:"
],
"metadata": {}
},
{
"outputs": [
{
"output_type": "execute_result",
"data": {
"text/plain": "Plot{Plots.GRBackend() n=3}",
"image/png": 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},
"metadata": {},
"execution_count": 25
}
],
"cell_type": "code",
"source": [
"plot(saturated_dependence(sparse_solver))"
],
"metadata": {},
"execution_count": 25
},
{
"cell_type": "markdown",
"source": [
"As we can see, for the sparse one, the standard monomials are \"trivial\"\n",
"because they are trivially detected as independent since they are not part of the basis.\n",
"The border is missing but the multiplication matrix for `y` can be computed first\n",
"and then, using this multiplication matrix, the missing relations for the border\n",
"can be obtained and then the multiplication matrix for `x` can be obtained."
],
"metadata": {}
},
{
"cell_type": "markdown",
"source": [
"---\n",
"\n",
"*This notebook was generated using [Literate.jl](https://github.com/fredrikekre/Literate.jl).*"
],
"metadata": {}
}
],
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