Make program act on itself by entering certain areas of code (as a string) which it must itself execute

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Apply a map to all coefficients of a polynomial

To apply a map to all coefficients of a polynomial, use map_coefficients.

For example, say we want to add two to each coefficient.

The function $x \mapsto x + 2$ is denoted in Python by lambda x: x + 2.

sage: K.<x_00, x_10, x_01, x_11> = QQ []
sage: g = K.random_element()
sage: g
7/18*x_00*x_01 + 51*x_00*x_11 - 126*x_01*x_11 - x_10
sage: g.map_coefficients(lambda x: x + 2)
43/18*x_00*x_01 + 53*x_00*x_11 - 124*x_01*x_11 + x_10

In this case, 7/18 became 43/18, 51 became 53, -126 became -124, and -1 became +1.

Note that this works around the question.

Accessing monomial $x_{ij}$ for given $i$ and $j$

For this, use string formatting. For instance:

sage: i = 0
sage: j = 1
sage: K(f'x_{i}{j}')
x_01