57539: Fixed creation unnamed table when using public schema

Problem:
- Table on public schema being lost when tried to be created

Solution:
- Used db_schema_name argument to specify schema name of adbc_ingest

pandas/io/sql.py

@@ -2380,7 +2380,9 @@ def to_sql(
             raise ValueError("datatypes not supported") from exc
 
         with self.con.cursor() as cur:
-            total_inserted = cur.adbc_ingest(table_name, tbl, mode=mode)
+            total_inserted = cur.adbc_ingest(
+                table_name=name, data=tbl, mode=mode, db_schema_name=schema
+            )
 
         self.con.commit()
         return total_inserted

pandas/tests/io/test_sql.py

@@ -1275,7 +1275,6 @@ def test_read_procedure(conn, request):
     res2 = sql.read_sql("CALL get_testdb();", conn)
     tm.assert_frame_equal(df, res2)
 
-
 @pytest.mark.parametrize("conn", postgresql_connectable)
 @pytest.mark.parametrize("expected_count", [2, "Success!"])
 def test_copy_from_callable_insertion_method(conn, expected_count, request):
@@ -1373,6 +1372,47 @@ def insert_on_conflict(table, conn, keys, data_iter):
         pandasSQL.drop_table("test_insert_conflict")
 
 
+@pytest.mark.parametrize("conn", postgresql_connectable)
+def test_to_sql_on_public_schema_postgres(conn, request):
+    # GH 15988: Example in to_sql docstring
+    conn = request.getfixturevalue(conn)
+
+    from sqlalchemy.engine import Engine
+    from sqlalchemy.sql import text
+
+    create_sql = text(
+        """
+    CREATE TABLE test_public_schema (
+        a  integer PRIMARY KEY,
+        b  numeric,
+        c  text
+    );
+    """
+    )
+    if isinstance(conn, Engine):
+        with conn.connect() as con:
+            with con.begin():
+                con.execute(create_sql)
+    else:
+        with conn.begin():
+            conn.execute(create_sql)
+
+    test_data = DataFrame([[1, 2.1, "a"], [2, 3.1, "b"]], columns=list("abc"))
+    test_data.to_sql(
+        name="test_public_schema",
+        con=conn,
+        if_exists="append",
+        index=False,
+        schema="public"
+    )
+
+    df_out = sql.read_sql_table("test_public_schema", conn, schema="public")
+    assert test_data.equals(df_out)
+
+    # Cleanup
+    with sql.SQLDatabase(conn, need_transaction=True) as pandasSQL:
+        pandasSQL.drop_table("test_public_schema")
+
 @pytest.mark.parametrize("conn", mysql_connectable)
 def test_insertion_method_on_conflict_update(conn, request):
     # GH 14553: Example in to_sql docstring