Updating README and Example Jupyter notebook with an example of usage…

… of point_on_variety function. Exposing new classes at module level via __init__. Forcing primary decomposition to return non-short Singular syntax for variables and exponents.

README.md

@@ -44,7 +44,69 @@ pytest --cov syngular/ --cov-report html tests/ --verbose
 
 ## Quick Start
 
+Define an ideal over a ring in two variables
 ```
 from syngular import Ideal, Ring
 I = Ideal(Ring('0', ('x1', 'x2'), 'dp'), ['x1*x2'])
 ```
+You can now inspect `I` to see what methods and attributes are available.
+
+## Solving arbitrary systems of polynomial equations
+
+Generate a $p$-adic solution to a system of 2 polynomial equations in 3 variables, controlling the precision to which they are solved.
+```
+field = Field("padic", 2 ** 31 - 1, 8)
+ring = Ring('0', ('x', 'y', 'z', ), 'dp')
+I = Ideal(ring, ['x*y^2+y^3-z^2, x^3+y^3-z^2', ])
+```
+
+The variety associated to `I` has 3 branches. In other words, the system of equations has 3 types of solutions.
+```
+(Q1, P1), (Q2, P2), (Q3, P3) = I.primary_decomposition
+```
+
+Generate a solution on the first branch
+```
+numerical_point = Q1.point_on_variety(field=field, directions=I.generators, valuations=(1, 1, ), ) 
+```
+is a dictionary of numerical values for each variable in the ring.
+
+These are small with valuations (1, 1)
+```
+list(map(lambda string: Polynomial(string, field).subs(numerical_point), Q1.generators))
+```
+
+while these are O(1) with valuations (0, 0)
+```
+list(map(lambda string: Polynomial(string, field).subs(numerical_point), Q2.generators))
+```
+
+See [arXiv:2207.10125](https://arxiv.org/pdf/2207.10125) Fig. 1 for a graphical depiction.
+
+## Citation
+
+If you found this library useful, please consider citing it and [Singular](https://www.singular.uni-kl.de/)
+
+
+```bibtex
+@inproceedings{DeLaurentis:2023qhd,
+    author = "De Laurentis, Giuseppe",
+    title = "{Lips: $p$-adic and singular phase space}",
+    booktitle = "{21th International Workshop on Advanced Computing and Analysis Techniques in Physics Research}: {AI meets Reality}",
+    eprint = "2305.14075",
+    archivePrefix = "arXiv",
+    primaryClass = "hep-th",
+    reportNumber = "PSI-PR-23-14",
+    month = "5",
+    year = "2023"
+}
+```
+
+```bibtex
+@misc {DGPS,
+ title = {{\sc Singular} {4-3-0} --- {A} computer algebra system for polynomial computations},
+ author = {Decker, Wolfram and Greuel, Gert-Martin and Pfister, Gerhard and Sch\"onemann, Hans},
+ year = {2022},
+ howpublished = {\url{http://www.singular.uni-kl.de}},
+}
+```

examples/Examples.ipynb

@@ -8,7 +8,7 @@
    "source": [
     "import sympy\n",
     "\n",
-    "from syngular import Ideal, Ring, QuotientRing, SingularException"
+    "from syngular import Ideal, Ring, QuotientRing, SingularException, Field, Polynomial"
    ]
   },
   {
@@ -255,11 +255,45 @@
     "L = I.reduce(J)\n",
     "assert L == Ideal(ring, ['0'])"
    ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# point on variety\n",
+    "field = Field(\"padic\", 2 ** 31 - 1, 8)\n",
+    "ring = Ring('0', ('x', 'y', 'z', ), 'dp')\n",
+    "I = Ideal(ring, ['x*y^2+y^3-z^2, x^3+y^3-z^2', ])\n",
+    "(Q1, P1), (Q2, P2), (Q3, P3) = I.primary_decomposition  # the variety associated to I has 3 branches. In other words, the system of equations has 3 types of solutions.\n",
+    "numerical_point = Q1.point_on_variety(field=field, directions=I.generators, valuations=(1, 1, ), )  # generate a solution on the first branch"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# these are small with valuations (1, 1)\n",
+    "list(map(lambda string: Polynomial(string, field).subs(numerical_point), Q1.generators))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# these are O(1) with valuations (0, 0)\n",
+    "list(map(lambda string: Polynomial(string, field).subs(numerical_point), Q2.generators))"
+   ]
   }
  ],
  "metadata": {
   "kernelspec": {
-   "display_name": "Python 3",
+   "display_name": "Python 3 (ipykernel)",
    "language": "python",
    "name": "python3"
   },
@@ -273,7 +307,7 @@
    "name": "python",
    "nbconvert_exporter": "python",
    "pygments_lexer": "ipython3",
-   "version": "3.8.10"
+   "version": "3.10.12"
   },
   "toc": {
    "base_numbering": 1,

syngular/__init__.py

@@ -8,3 +8,4 @@
 from .qring import QuotientRing, QRing  # noqa
 from .tools import SingularException  # noqa
 from .field import Field  # noqa
+from .polynomial import Polynomial  # noqa

syngular/ideal.py

@@ -154,6 +154,7 @@ def primary_decomposition(self):
                              f"ideal i = {self};",
                              # f"ideal gb = {','.join(self.groebner_basis)};",
                              "def pr = primdecGTZ(i);",   # options: GTZ / SY
+                             "short=0;",
                              "print(pr);",
                              "$"]
         output = execute_singular_command(singular_commands)