Count the number of polynomial roots between 0 and 10:

Count roots of a polynomial in a closed rectangle:

Count roots of a real elementary function in a real interval:

Count roots of a holomorphic function in a closed rectangle:

The number of 17 roots of unity in the closed unit square in the first quadrant:

Roots on the boundary are counted:

Check that a function has exactly one root in an interval:

Use FindRoot to approximate the root:

Compute a contour integral of logarithmic derivative of a function using the formula , where is the number of roots for a holomorphic function :

Compare with the result of numeric integration:

Test stability of equilibria at 0 of linear dynamical systems by counting the roots of the CharacteristicPolynomial[m,x] in the right half-plane:

Use a bound for max absolute value of the roots:

Count the roots in the right half-plane:

Since all eigenvalues of m have negative real parts, the equilibrium is asymptotically stable:

Try a different matrix:

This time there are roots with non-negative real parts:

The equilibrium is not asymptotically stable:

The number of complex roots of a polynomial is equal to its degree:

This gives a bound on absolute values of roots of a polynomial:

The polynomial indeed has 10 roots within the Cauchy bounded region:

The number of real roots of a polynomial with nonzero terms is at most :

This polynomial has the maximal possible number of real roots:

Use Reduce to find polynomial roots:

Count roots:

Find roots:

Use RootIntervals to find isolating intervals for roots:

Count the real roots:

Isolate the real roots:

Use NumberFieldSignature to count the real roots and the pairs of complex roots of a polynomial:

Roots at the endpoints of the interval are included:

Count the roots of a meromorphic function:

Plot the roots as spikes: