Should I contact publishing company before publishing my poem inspired by their book?

You need to consult with a lawyer who is experienced with copyright and publishing. Having a lawyer on board as part of your writing and publishing work is crucial.

Any derivative work requires a license from the copyright holder.

In that case, your poem could be considered a derivative work, thus the book's author could actually own the copyright on your poem, rather than yourself. You would most likely have to license the rights from the author/publisher of the book in order to legally publish your poem.

If the original book was written and/or published prior to approximately 1925, it is probably in the public domain and you would not need any permission to publish your poem. If the book was written since then, it is probably still copyrighted and you would probably need to get permission to release the poem. However, if your poem is a literary criticism of the original work, a parody of it, or any other use falling under the Fair Use defense, you don't need permission. Be aware also, that certain characters may be separately protected beyond the book. Walt Disney characters are a good example of this.