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• Lists of real numbers
  Use Sage syntax to enter a number, a list of numbers, or a dictionary of the form {param: number}. The lists and dictionaries may be nested. The numbers might be given via formulas involving standard functions such as sin and sqrt, as well as standard constants such as e and pi.
  Examples:
  • -2, pi, e searches for certain constants.
  • {n: 2^n for n in [1..10]} searches for $2^1$, ..., $2^{10}$.
  • RIF(10,11) searches for all numbers contained in the interval $[10,11]$.
  • {n: sin(RIF(pi)/n) for n in [1..10]} searches for $\sin(\pi/n)$ using real-interval arithmetic, without symbolic simplification.
  • {a: {b: a/b for b in [1..5] if gcd(a,b) == 1} for a in [-5..5]} searches certain small rational numbers.
• Lists of complex numbers
  Similarly one can search for complex numbers. Recommended parents are CIF (complex interval field), and CBF (complex ball field), although CC (complex field via floats) should work as well unless too much precision is lost during the computation.
• Lists of p-adic numbers
  Similarly one can search for p-adic numbers in $\mathbb{Z}_p$ and $\mathbb{Q}_p$.
  Examples:
  • [field(5) for field in [Qp(2,prec=4)]] searches for $5 + O(2^4)$ = $2^0 + 2^2 + O(2^4)$, that is, any $2$-adic integer with first four $2$-adic digits $1$, $0$, $1$, $0$.
  • [{n: field(n) for n in [1..10]} for field in [Qp(2)]] searches $1$, ..., $10$ as $2$-adic numbers with SageMath's default relative precision of 20 $2$-adic digits. For even integers, the absolute precision is higher because their valuation is positive.
• Lists of polynomials over $\mathbb{Q}$
  Similarly one can search for multivariate polynomials over $\mathbb{Q}$. Only expressions are accepted. Bind a polynomial ring inside a comprehension and supply its coefficient list, starting with the constant coefficient:
  
  • [{n: ring([n,0,1]) for n in [1..10]} for ring in [PolynomialRing(QQ,'x')]] searches for polynomials $x^2+n$ for integers $n=1,\ldots,10$.

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